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RESOU RCE OPTIONS • RESOU RCE OPTIONS • RESOU RCE OPTIONS • RESOU RCE
Exponents and Polynomials
Section 7A
Section 7B
Exponents
Polynomials
7-1 Integer Exponents
7-6 Polynomials
7-2 Powers of 10 and Scientific Notation
7-7 Algebra Lab Model Polynomial Addition and Subtraction
7-3 Algebra Lab Explore Properties of Exponents
7-3 Multiplication Properties of Exponents
7-8 Algebra Lab Model Polynomial Multiplication
7-4 Division Properties of Exponents
7-8 Multiplying Polynomials
7-5 Rational Exponents
Connecting Algebra to Geometry Volume and Surface Area
7-9 Special Products of Binomials
Pacing Guide for 45-Minute Classes
Calendar Planner®
Chapter 7
Countdown Weeks 15 , 16 , 17
DAY 1
DAY 2
DAY 3
DAY 4
DAY 5
7-1 Lesson
7-2 Lesson
7-3 Algebra Lab
7-3 Lesson
7-4 Lesson
DAY 6
DAY 7
DAY 8
DAY 9
DAY 10
7-5 Lesson
Multi-Step Test Prep
7-6 Lesson
7-7 Algebra Lab
7-7 Lesson
7-7 Lesson
7-8 Algebra Lab
DAY 11
DAY 12
DAY 13
DAY 14
DAY 15
7-8 Lesson
Connecting Algebra
to Geometry
7-9 Lesson
Multi-Step Test Prep
Chapter 7 Review
DAY 16
Chapter 7 Test
Pacing Guide for 90-Minute Classes
Calendar Planner®
Chapter 7
DAY 1
DAY 2
DAY 3
DAY 4
DAY 5
Chapter 6 Test
7-1 Lesson
7-2 Lesson
7-3 Algebra Lab
7-3 Lesson
7-4 Lesson
7-5 Lesson
Multi-Step Test Prep
7-6 Lesson
7-7 Algebra Lab
7-7 Lesson
DAY 6
DAY 7
DAY 8
DAY 9
7-7 Lesson
7-8 Algebra Lab
7-8 Lesson
Connecting Algebra
to Geometry
7-9 Lesson
Multi-Step Test Prep
Chapter 7 Review
Chapter 7 Test
8-1 Lesson
456A
Chapter 7
E OPTIONS • RESOU RCE OPTIONS • RESOU RCE OPTIONS • RESOU RCE OPTIONS
DIAGNOSE
Assess
Prior
Knowledge
PRESCRIBE
Before Chapter 7
Prescribe intervention.
Are You Ready? SE p. 457
Before Every Lesson
Prescribe intervention.
Warm Up TE
Reteach CRB
During Every Lesson
Diagnose understanding of lesson concepts.
Prescribe intervention.
Check It Out! SE
Questioning Strategies TE
Think and Discuss SE
Journal TE
Success for Every Learner
Lesson Tutorial Videos
After Every Lesson
Formative
Assessment
Diagnose mastery of lesson concepts.
Prescribe intervention.
Lesson Quiz TE
Test Prep SE
Test and Practice Generator
Reteach CRB
Test Prep Doctor TE
Homework Help Online
Before Chapter 7 Testing
Diagnose mastery of concepts in chapter.
Prescribe intervention.
Ready to Go On? SE pp. 495, 529
Multi-Step Test Prep SE pp. 494, 528
Section Quizzes AR
Test and Practice Generator
Scaffolding Questions TE pp. 494, 528
Reteach CRB
Lesson Tutorial Videos
Before High Stakes Testing
Diagnose mastery of benchmark concepts.
Prescribe intervention.
College Entrance Exam Practice SE p. 535
Standardized Test Prep SE pp. 538–539
College Entrance Exam Practice
After Chapter 7
Summative
Assessment
KEY:
SE = Student Edition
Check mastery of chapter concepts.
Prescribe intervention.
Multiple-Choice Tests (Forms A, B, C)
Free-Response Tests (Forms A, B, C)
Performance Assessment AR
Cumulative Test AR
Test and Practice Generator
Reteach CRB
Lesson Tutorial Videos
TE = Teacher’s Edition
CRB = Chapter Resource Book AR = Assessment Resources
Available online
Available on CD- or DVD-ROM
456B
RESOU RCE OPTIONS • RESOU RCE OPTIONS • RESOU RCE OPTIONS • RESOU RCE
E
Lesson Resources
Before the Lesson
Practice the Lesson
Prepare Teacher One Stop JG8E@J?
Practice Chapter Resources
• Editable lesson plans
• Calendar Planner
• Practice A, B, C
Practice and Problem Solving Workbook JG8E@J?
IDEA Works!® Modified Worksheets and Tests
ExamView Test and Practice Generator
Lesson Transparencies
• Teacher Tools
JG8E@J?
Homework Help Online
JG8E@J?
Online Interactivities
Interactive Online Edition
Teach the Lesson
• Homework Help
Introduce Alternate Openers: Explorations
Lesson Transparencies
Apply Chapter Resources
• Warm Up
• Problem of the Day
• Problem Solving JG8E@J?
Practice and Problem Solving Workbook JG8E@J?
Interactive Online Edition
Teach Lesson Transparencies
• Chapter Project
• Teaching Transparencies
Project Teacher Support
Know-It Notebook™
• Vocabulary
• Key Concepts
Power Presentations
Lesson Tutorial Videos
Interactive Online Edition
After the Lesson
JG8E@J?
Reteach Chapter Resources
• Reteach
• Lesson Activities
• Lesson Tutorial Videos
Lab Activities
Lab Resources Online
Online Interactivities
TechKeys
Success for Every Learner
Solutions Key
Know-It Notebook™
JG8E@J?
• Big Ideas
• Chapter Review
Extend Chapter Resources
• Challenge
Technology Highlights for the Teacher
Power Presentations
Teacher One Stop
Dynamic presentations to engage students.
Complete PowerPoint® presentations for
every lesson in Chapter 7.
KEY:
SE = Student Edition
456C
Chapter 7
TE = Teacher’s Edition
%,,
JG8E@J?
assessments. Includes lesson planning, test
generation, and puzzle creation software.
English Language Learners
JG8E@J? Spanish version available
Premier Online Edition
JG8E@J?
Chapter 7 includes Tutorial Videos, Lesson
Activities, Lesson Quizzes, Homework Help,
and Chapter Project.
Available online
Available on CD- or DVD-ROM
E OPTIONS • RESOU RCE OPTIONS • RESOU RCE OPTIONS • RESOU RCE OPTIONS
Reaching All Learners
Teaching tips to help all learners appear throughout the chapter. A few that target specific students are included in the lists below.
All Learners
On-Level Learners
Lab Activities
Practice and Problem Solving Workbook JG8E@J?
Know-It Notebook
Practice B ............................................................................. CRB
Problem Solving .................................................................. CRB
Vocabulary Connections ..............................................SE p. 458
Questioning Strategies ...........................................................TE
Ready to Go On? Intervention JG8E@J?
Know-It Notebook
Homework Help Online
JG8E@J?
Online Interactivities
Special Needs Students
Practice A ............................................................................. CRB
Reteach................................................................................. CRB
Inclusion .....................................................TE pp. 461, 505, 513
IDEA Works!® Modified Worksheets and Tests
Ready to Go On? Intervention JG8E@J?
Know-It Notebook
JG8E@J?
Online Interactivities
JG8E@J?
Lesson Tutorial Videos
Practice C ............................................................................. CRB
Challenge ............................................................................. CRB
Challenge Exercises ...............................................................SE
Reading and Writing Math Extend ..............................TE p. 459
Critical Thinking ..........................................................TE p. 515
Developing Learners
Practice A ............................................................................. CRB
Reteach................................................................................. CRB
Vocabulary Connections ..............................................SE p. 458
Questioning Strategies ...........................................................TE
Ready to Go On? Intervention JG8E@J?
Know-It Notebook
Homework Help Online
JG8E@J?
Online Interactivities
JG8E@J?
Lesson Tutorial Videos
Ready To Go On? Enrichment JG8E@J?
English Language Learners
ENGLISH
LANGUAGE
LEARNERS
Are You Ready? Vocabulary ........................................SE p. 457
Vocabulary Connections ..............................................SE p. 458
Vocabulary Review ......................................................SE p. 530
English Language Learners................................TE pp. 459, 476
Success for Every Learner
Know-It Notebook
Multilingual Glossary
JG8E@J?
Lesson Tutorial Videos
Technology Highlights for Reaching All Learners
Lesson Tutorial Videos
JG8E@J?
Starring Holt authors Ed Burger and Freddie
Renfro! Live tutorials to support every
lesson in Chapter 7.
KEY:
SE = Student Edition TE = Teacher’s Edition
Multilingual Glossary
Searchable glossary includes definitions
in English, Spanish, Vietnamese, Chinese,
Hmong, Korean, and 4 other languages.
CRB = Chapter Resource Book
Online Interactivities
Interactive tutorials provide visually engaging
alternative opportunities to learn concepts and
master skills.
JG8E@J? Spanish version available
Available online
Available on CD- or DVD-ROM
456D
RESOU RCE OPTIONS • RESOU RCE OPTIONS • RESOU RCE OPTIONS • RESOU RCE
E
Ongoing Assessment
Assessing Prior Knowledge
Lesson Assessment
Determine whether students have the prerequisite concepts
and skills for success in Chapter 7.
Provide formative assessment for each lesson of Chapter 7.
............................SE p. 457
Warm Up
.................................................................TE
Test Preparation
Provide review and practice for Chapter 7 and standardized
tests.
Multi-Step Test Prep ..........................................SE pp. 494, 528
Study Guide: Review .........................................SE pp. 530–533
Test Tackler ........................................................SE pp. 536–537
Standardized Test Prep......................................SE pp. 538–539
College Entrance Exam Practice ..................................SE p. 535
Countdown to Testing
........................ SE pp. C4–C27
®
IDEA Works! Modified Worksheets and Tests
Questioning Strategies ...........................................................TE
Think and Discuss ...................................................................SE
Check It Out! Exercises ...........................................................SE
Journal ....................................................................................TE
Lesson Quiz
.............................................................TE
Alternative Assessment ..........................................................TE
IDEA Works!® Modified Worksheets and Tests
Weekly Assessment
Provide formative assessment for each section of Chapter 7.
Multi-Step Test Prep ..........................................SE pp. 494, 528
JG8E@J? .................SE pp. 495, 529
Section Quizzes JG8E@J? ........................................................ AR
Test and Practice Generator JG8E@J?
.... Teacher One Stop
Alternative Assessment
Assess students’ understanding of Chapter 7 concepts
and combined problem-solving skills.
Chapter Assessment
Chapter 7 Project .........................................................SE p. 456
Alternative Assessment ..........................................................TE
Performance Assessment JG8E@J? ........................................ AR
Portfolio Assessment JG8E@J? ............................................... AR
Provide summative assessment of Chapter 7 mastery.
Chapter 7 Test ..............................................................SE p. 534
Chapter Test (Levels A, B, C) JG8E@J? ................................... AR
• Multiple Choice
• Free Response
Cumulative Test JG8E@J? ....................................................... AR
Test and Practice Generator JG8E@J?
.... Teacher One Stop
®
IDEA Works! Modified Worksheets and Tests
Technology Highlights for Assessment
JG8E@J?
intervention for Chapter 7 prerequisite skills.
KEY:
456E
SE = Student Edition
Chapter 7
TE = Teacher’s Edition
Automatically assess understanding
of and prescribe intervention for
Sections 7A and 7B.
AR = Assessment Resources
JG8E@J? Spanish version available
Test and Practice Generator
JG8E@J?
Use Chapter 7 problem banks to create
assessments and worksheets to print out
or deliver online. Includes dynamic problems.
Available online
Available on CD- or DVD-ROM
E OPTIONS • RESOU RCE OPTIONS • RESOU RCE OPTIONS • RESOU RCE OPTIONS
Formal Assessment
Three levels (A, B, C) of multiple-choice and free-response chapter tests, along
with a performance assessment, are available in the Assessment Resources.
A Chapter 7 Test
A Chapter 7 Test
C Chapter 7 Test
C Chapter 7 Test
MULTIPLE CHOICE
FREE RESPONSE
PERFORMANCE ASSESSMENT
B Chapter 7 Test
B Chapter 7 Test
Chapter 7 Test
9. Simplify (3a5)2.
1. Which of the following is equivalent to 2−3?
A (−2)(−2)(−2)
B −
C
1
(2)(2)(2)
1
(2)(2)(2)
D (2)(2)(2)
2. Evaluate (3 + x)−2 for x = −1.
F −4
G
H
1
4
10
9
1
4b3
1
64b3
D
4
b3
F 2.67 × 106
H 26.7 × 104
7
5
F 10−5
H 104
G 10−4
J 105
5. In 2000, the population of Texas was
population in scientific notation.
A 2.09 × 105
C 20.9 × 106
B 2.09 × 107
D 209 × 105
H 0.062
1
620
J 6200
a
.
a2
________________________________________
A a4
C a10
B a6
D a16
4. Write 0.0000001 as a power of 10.
20
F
2y
x3
H
G
2y 20
x15
J 32y5
32y
x15
5. In 2000, the population of Indiana was
population in scientific notation.
−2
11.
.
________________________________________
⎛ 4y 6 ⎞
12. ⎜ 3 ⎟
⎝ x ⎠
6. Write 8.5 × 10−7 in standard form.
b8
25
A
1
25b 4
C
B
5
b6
D −5
________________________________________
2
2
________________________________________
⎛ 0⎞
13. ⎜ 3 ⎟
⎝b ⎠
H x9
J x18
7. y7 • y3
________________________________________
________________________________________
A 4
2
A 2x + 2x − 4
C 6
B 5
B 9x2 − 5x − 4
D 64
14. Simplify the quotient
(6.4 × 109) ÷ (1.6 × 10−3). Write the
C 9x2 − 2x − 7
2
________________________________________
4
F 2
H 16
G 8
22. Subtract (7a2 − 3a) − (5a2 − 5a).
J 512
3
2
C x
B x3
Simplify.
D 9x − 5x − 4
17. Simplify (x 2 )4 x 3 . All variables
represent nonnegative numbers.
2(x)
D x9
18. When written in standard form, which
polynomial has a leading coefficient of 5?
F −7 + 6y + 5y2
G x+5
H x2 − 5x3 + 2x
2
J 5y + 3y − 4
19. Classify the polynomial 3x5 + 3 according
to its degree and number of terms.
A cubic binomial
B cubic trinomial
F 2a2 − 8a
H 4
G 2a2 + 2a
J 12a2 − 8a
F 3w2 − 2
H 4w − 2
G 3w2 − 2w
J 8w − 4
25. Multiply (x − 5)(2x + 4).
C 2x2 − 20
A −6x
2
2
B 2x − 26
D 2x − 6x − 20
G b3 − 2b2 − 22b − 21
J 464 ft2
________________________________________
17. Simplify (x 3 )3 5 y 5 . All variables
D quintic trinomial
G 336 ft2
Converts between standard notation and engineering notation.
______
Compares and contrasts scientific and engineering notation.
Scoring Rubric
Level 1: Student is not able to solve any of the problems.
Chapter 7 Test
(continued)
21. Add (5x2 − 2x + 9) + (2x 2 − 4).
________________________________________
22. Subtract (10a2 − 6a) − (7a2 − 8a).
23. Multiply (−5rs4)(3r 5s2).
24. A rectangle has width w and its length is
2 units shorter than 3 times the width, or
3w − 2. Write a polynomial for the area of
the rectangle.
F b3 − 5b2 − 21
H 384 ft2
Uses powers of 10 to represent periods.
______
Commas are used to separate a large number into groups of three digits,
or periods. Decimal numbers less than 1 can be written with spaces to
distinguish periods. Periods provide a shorthand way to write the numbers.
360
,
millions
period
000
,
thousands
period
000
= 360 million
H 3b3 − 15b2 − 21b
2
J 4b − 20b − 28b
2
27. Multiply (2x + 7) .
A 2x2 + 7
C 4x2 + 14x + 49
B 4x2 + 49
D 4x2 + 28x + 49
28. Which product results in x2 − 100?
F (x − 10)2
G (x + 10)2
H x(x − 100)
0. 000
,
units
period
000
,
thousandths
period
millionsths
period
045
= 45 billionths
billionths
period
Write these numbers using periods (trillion, billion, million, thousand,
thousandths, millionths, billionths, and trillionths).
1. 35,000,000,000,000
26. Multiply (b + 3)(b2 − 5b − 7).
F 114 ft2
1
15. 32 5
16.
3
Rewrites numbers using words to describe periods.
______
________________________________________
23. Multiply (4rs3)(6r 2s3).
A 10r 2s9
C 24r 2s9
B 10r 3s6
D 24r 3s6
C quintic binomial
20. Brett has 100 feet of fence with which to
make a rectangular cage for his dog. The
area of the cage in square feet is given
by the polynomial −w2 + 50w, where w is
the width of the cage in feet. What is the
area of the cage if the width is 8 feet?
As a class, go over the two introductory examples. Mention that the use
of the comma, space, and decimal point to separate periods is a regional,
North American convention. Around the world, there are a variety of
symbols used to denote separators in multi−digit numbers.
Level 3: Student solves most problems correctly but gives a faulty
comparison between scientific and engineering notation.
B Chapter 7 Test
(continued)
21. Add (2x2 − 5x − 7) + (7x 2 + 3).
16. Simplify 64 3 .
Many scientific and graphing calculators have both a scientific notation
mode (SCI) and an engineering notation mode (ENG). If you routinely
use calculators in your class, you may want to allow students to use them
on this activity. Calculators can be particularly beneficial for problem 16
because students can enter and compare numbers under both notations.
Level 2: Student solves a few problems and gives no comparison.
B Chapter 7 Test
(continued)
1
Overview
Level 4: Student solves all problems correctly and gives a solid
comparison between scientific and engineering notation.
________________________________________
15. Simplify 256 4 .
Preparation Hints
______
Simplify.
J 4.8 × 10
F x2
Individuals
Performance Indicators
−2
−24
G x3
Grouping
8. (x20)4
H 3 × 10−3
30−40 minutes
Students first rewrite numbers using periods. Then they write the periods
using powers of 10, which have exponents that are always multiples of 3.
Students then use engineering notation, a variation of scientific notation,
to rewrite numbers.
a16
a4
________________________________________
G 3 × 10−24
D y50
8. Simplify (x ) .
A x
Simplify.
________________________________________
20
Time
Review powers of 10 and scientific notation. Review the place-value names
of digits in decimal numbers and define groups of digits as periods.
3. Simplify 5b−8.
F 1.728 × 10−12
6 3
1
10. A computer modem can transmit
1.5 × 106 bytes per second. How many
bytes can it transmit in 300 seconds?
________________________________________
14. Simplify the quotient
(7.2 × 10−18) ÷ (2.4 × 106).
C y15
B y5
________________________________________
8
⎛ 5⎞
13. Simplify ⎜ 4 ⎟
⎝b ⎠
7. Simplify y10 • y5.
A y2
________________________________________
⎛ 2y 4 ⎞
12. Simplify ⎜ 3 ⎟ .
⎝ x ⎠
6. Which of the following is the standard
form of 6.2 × 10−2?
F −620
J 80.1 × 10
This performance task assesses the student’s ability to use powers of 10 to
convert between standard, period, engineering, and scientific notations.
________________________________________
2. Evaluate (5 − x)−2 for x = −1.
Purpose
9. (2a3)5
5
4. Which power of 10 is equivalent to
0.00001?
G
D 9a10
G 8.01 × 10
3. Simplify 4b−3.
B
C 9a5
B 3a10
11. Simplify
C
1. Simplify 4 .
A 3a7
10. In June 2005, Miami International Airport
served about 8.9 × 104 airline passengers
per day. Find the approximate number of
airline passengers served in total during
the 30 days of June.
J 10
A −64b3
−3
4
27 3
________________________________________
________________________________________
24. A rectangle has width w and its length is
4 units longer than 2 times the width, or
2w + 4. Write a polynomial for the area of
the rectangle.
1
represent nonnegative numbers.
________________________________________
18. When the polynomial 5x − 2x3 + 8x2 − 7
is written in standard form, what is the
________________________________________
20. Genie has 100 feet of fence with which
to make a rectangular cage for her
rabbit. If she uses the wall of her house
as one side, the area of the cage in
square feet is given by the polynomial
−2w2 + 100w, where w is the width of
the cage in feet. What is the area of the
cage if the width is 15 feet?
4. 0.000 012
________________________
________________________________________
Multiply.
25. (x + 6)(3x − 8)
________________________
10. 1 thousand
________________________
________________________________________
2. 425,000
3. 9,500,000
________________________
5. 0.000 000 000 005
________________________
________________________
6. 0.069
________________________
Write these numbers as powers of 10.
7. 1 trillion
8. 1 billion
9. 1 million
________________________
11. 1 thousandth
________________________
12. 1 millionth
________________________
________________________
Engineering notation is a method of writing a number with a power of 10
to emphasize the period of the number. It helps to write the period first.
2
26. (b − 4)(b + 3b − 2)
________________________________________
19. Classify the polynomial 5x2 + 9x + 1
according to its degree and number of
terms.
________________________
________________________________________
27. (3x − 4)2
________________________________________
28. (2x + 4)(2x − 4)
________________________________________
0.000 000 045 = 45 billionths = 45 109
Write these numbers in engineering notation.
13. 96,000,000,000
________________________
14. 108,000
15. 0.000 004
________________________
715, complete this table to compare
and contrast scientific notation and
engineering notation.
Notation
________________________
Scientific
Engineering
17. Convert 5.8 105 to engineering notation.
______________________________
________________________________________
J (x + 10)(x − 10)
JG8E@J?
Test & Practice Generator
Modified chapter tests that address special
learning needs are available in IDEA Works!®
Modified Worksheets and Tests.
Create and customize Chapter 7 Tests. Instantly
generate multiple test versions, answer keys, and
Spanish versions of test items.
456F
Exponents and
Polynomials
SECTION
7A Exponents
7A
7-1
Exponents
On page 494, students write, solve,
and graph equations
to model real-world
speed-of-light situations.
Exercises designed to prepare
students for success on
the Multi-Step Test Prep
can be found in each
lesson.
SECTION
7B
7-2 Powers of 10 and Scientific
Notation
Lab
Explore Properties of
Exponents
7-3
Multiplication Properties of
Exponents
7-4 Division Properties of
Exponents
7-5
Rational Exponents
7B Polynomials
7-6
Polynomials
Lab
and Subtraction
Polynomials
On page 528,
students multiply
polynomials to
model a real-world
Integer Exponents
Polynomials
Lab
Model Polynomial
Multiplication
7-8
Multiplying Polynomials
7-9 Special Products of Binomials
area situation.
Exercises designed to prepare
students for success on
the Multi-Step Test Prep
can be found in each
lesson.
• Use exponents and scientific notation
to describe numbers.
• Use laws of exponents to simplify
monomials.
• Perform operations with polynomials.
Interactivities Online
▼
Every Second Counts
How many seconds until you
large numbers such as this one.
KEYWORD: MA7 ChProj
Lessons 7-2, 7-5, 7-7
Lesson Tutorials Online
456
Chapter 7
Every Second Counts
Project Resources
In the Chapter Project, students consider
the passage of time in seconds. First they
calculate large numbers of seconds, such
as how many seconds they’ve been alive or
how many seconds until graduation. Then
they calculate fractions of seconds as they
learn about a car’s braking distance and
driver reaction time. In each case, students
use exponents and scientific notation to
work with these very large and very small
numbers.
All project resources for teachers and
students are provided online.
A11NLS_c07_0456-0459.indd 456
Lesson Tutorial Videos are
available for EVERY example.
456
Chapter 7
Materials:
• calculators
KEYWORD: MA7 ProjectTS
8/18/09 9:50:17 AM
Vocabulary
Match each term on the left with a definition on the right.
1. Associative Property F A. a number that is raised to a power
Organizer
2. coefficient B
B. a number that is multiplied by a variable
3. Commutative
Property C
C. a property of addition and multiplication that states you can
add or multiply numbers in any order
4. exponent D
D. the number of times a base is used as a factor
5. like terms E
E. terms that contain the same variables raised to the same
powers
Objective: Assess students’
understanding of prerequisite skills.
F. a property of addition and multiplication that states you can
group the numbers in any order
Assessing Prior
Knowledge
Exponents
INTERVENTION
Write each expression using a base and an exponent.
6. 4 · 4 · 4 · 4 · 4 · 4 · 4 4 7
7. 5 · 5 5 2
9. x · x · x x 3
8. (-10)(-10)(-10)(-10) (-10)4
10. k · k · k · k · k k 5
11. 9 9 1
13. -12 2 -144
14. 5 3 125
Evaluate Powers
Evaluate each expression.
12. 3 4 81
5
16. 4 64
17. (-1) 1
19. 25.25 × 100 2525
20. 2.4 × 6.5 15.6
whether intervention is necessary
or whether enrichment is
appropriate.
Resources
Intervention and
Enrichment Worksheets
6
3
15. 2 32
Diagnose and Prescribe
Multiply Decimals
Multiply.
18. 0.006 × 10 0.06
Combine Like Terms
Simplify each expression.
21. 6 + 3p + 14 + 9p 20 + 12p
22. 8y - 4x + 2y + 7x - x 10y + 2x
23. (12 + 3w - 5) + 6w - 3 - 5w 4 + 4w
24. 6n - 14 + 5n 11n - 14
Squares and Square Roots
Tell whether each number is a perfect square. If so, identify its positive square root.
25. 42 no
26. 81 yes; 9
27. 36 yes; 6
28. 50 no
29. 100 yes; 10
30. 4 yes; 2
31. 1 yes; 1
32. 12 no
Exponents and Polynomials
NO
INTERVENE
A1NL11S_c07_0456-0459.indd 457
457
YES
Diagnose and Prescribe
ENRICH
6/25/09 9:05:06 AM
ARE YOU READY? Intervention, Chapter 7
Prerequisite Skill
Worksheets
CD-ROM
Exponents
Skill 7
Activity 7
Evaluate Powers
Skill 8
Activity 8
Multiply Decimals
Skill 45
Activity 45
Combine Like Terms
Skill 57
Activity 57
Squares and Square Roots
Skill 6
Activity 6
Online
Diagnose and
Prescribe Online
Enrichment, Chapter 7
Worksheets
CD-ROM
Online
457
CHAPTER
Study Guide:
Preview
7
Organizer
Key
Vocabulary/Vocabulario
Objective: Help students
Previously, you
GI
organize the new concepts they
will learn in Chapter 7.
<D
@<I
binomial
binomio
exponential expressions.
degree of a monomial
simplified algebraic
expressions by combining like
terms.
degree of a polynomial
coeficiente principal
monomial
monomio
perfect-square
trinomial
perfecto
polynomial
polinomio
scientific notation
notación científica
standard form of a
polynomial
forma estándar de un
polinomio
trinomial
trinomio
• wrote and evaluated
•
Online Edition
Multilingual Glossary
Resources
You will study
PuzzleView
• properties of exponents.
• powers of 10 and scientific
Multilingual Glossary Online
•
KEYWORD: MA7 Glossary
notation.
multiply polynomials by using
properties of exponents and
combining like terms.
Vocabulary Connections
Vocabulary Connections
To become familiar with some of the
vocabulary terms in the chapter, consider
the following. You may refer to the chapter,
the glossary, or a dictionary if you like.
1. a method of writing very large
and very small numbers
2. the coefficient of the first (lead)
term, which is the number
attached to the variable of that
term
You can use the skills
in this chapter
• to model area, perimeter, and
volume in geometry.
to express very small or very
large quantities in science
classes such as Chemistry,
Physics, and Biology.
3. an expression with exactly 2
terms
•
triangle, tripod; they all have
three of something; an expression with exactly three terms.
• in the real world to model
population growth or
decline.
1. Very large and very small numbers are
often encountered in the sciences. If
notation means a method of writing
something, what might scientific
notation mean?
2. A polynomial written in standard form
may have more than one algebraic term.
What do you think the leading
coefficient of a polynomial is?
3. A simple definition of monomial is “an
expression with exactly one term.” If the
prefix mono- means “one” and the prefix
bi- means “two,” define the word
binomial .
4. What words do you know that begin with
the prefix tri-? What do they all have in
common? Define the word trinomial
based on the prefix tri- and the
information given in Problem 3.
458
Chapter 7
A1NL11S_c07_0456-0459.indd 458
458
Chapter 7
6/25/09 9:05:26 AM
CHAPTER
7
Organizer
Objective: Help students apply
Follow this strategy when solving word problems.
strategies to understand and retain
key concepts.
• Read the problem through once.
• Identify exactly what the problem asks you to do.
GI
• Read the problem again, slowly and carefully, to break it into parts.
<D
@<I
Online Edition
• Highlight or underline the key information.
• Make a plan to solve the problem.
ENGLISH
LANGUAGE
LEARNERS
the Problem
From Lesson 6-6
29. Multi-Step Linda works at a pharmacy for \$15 an hour. She also baby-sits for
\$10 an hour. Linda needs to earn at least \$90 per week, but she does not want
to work more than 20 hours per week. Show and describe the number of hours
Linda could work at each job to meet her goals. List two possible solutions.
Step 1
Identify exactly what
to do.
Discuss Key information can include
more than just numbers. Words and
phrases such as not, no more than,
whole number solutions, inches, and
two different ways can be crucial to
correctly solving a problem.
• Show and describe the number of hours Linda
can work at each job and earn at least \$90
per week, without working more than 20 hours
per week.
Extend As you present word
problems in Chapter 7 to the class,
should be highlighted and why.
• List two possible solutions of the system.
Step 2
Step 3
Break the problem into
parts. Highlight or
underline the key
information.
Make a plan to solve
the problem.
• Linda has two jobs. She makes \$15 per hour at
one job and \$10 per hour at the other job.
• She wants to earn at least \$90 per week.
• She does not want to work more than 20 hours
per week.
1a. Find the length and width of
the rectangle.
• Write a system of inequalities.
• Solve the system.
b. The difference between length
and width is 14 units. The area
is 120 square units.
• Identify two possible solutions of the system.
c. Write and solve a system of
equations.
Try This
For the problem below,
a. identify exactly what the problem asks you to do.
b. break the problem into parts. Highlight or underline the key information.
c. make a plan to solve the problem.
1. The difference between the length and the width of a rectangle is 14 units. The
area is 120 square units. Write and solve a system of equations to determine
the length and the width of the rectangle. (Hint: The formula for the area of a
rectangle is A = w.)
Exponents and Polynomials
A1NL11S_c07_0456-0459.indd 459
Connection
459
6/25/09
Powers of Ten
by Philip and Phylis Morrison
This is a wondrous yet practical
guide to powers of 10 and exponential growth. The book begins
with one billion light years. In
9:05:34 AM
1
40 steps, each step depicting _
10
the scale of the previous step, it
moves to subatomic particles.
their own books using 1 ft
(10 0 ft) as step 6, 10 1 ft as step 7,
10 2 ft as step 8, and so on, up to
step 11. Then 10 -1 ft will be step
5, 10 -2 ft will be step 4, etc. For
each step, have students name
(and perhaps locate a photo of)
an object near that length.
459
SECTION
7A Exponents
One-Minute Section Planner
Lesson
Lab Resources
Lesson 7-1 Integer Exponents
•
•
□
Optional
Evaluate expressions containing zero and integer exponents.
Simplify expressions containing zero and integer exponents.
NAEP
✔ SAT-10
✔ ACT
✔ SAT
✔ SAT Subject Tests
□
□
□
graphing calculator
□
Lesson 7-2 Powers of 10 and Scientific Notation
•
•
□
Optional
Evaluate and multiply by powers of 10.
Convert between standard notation and scientific notation.
NAEP
SAT
SAT Subject Tests
✔ SAT-10
✔ ACT
□
□
□
Use patterns to explore multiplication properties of exponents.
SAT-10 ✔ NAEP
ACT
SAT
SAT Subject Tests
□
□
□
graphing calculator
□
7-3 Algebra Lab Explore Properties of Exponents
•
□
Materials
□
Algebra Lab Activities
7-3 Lab Recording Sheet
Lesson 7-3 Multiplication Properties of Exponents
• Use multiplication properties of exponents to evaluate and simplify
expressions.
✔ SAT-10 □
✔ NAEP □
✔ ACT
□
✔ SAT
□
✔ SAT Subject Tests
□
Lesson 7-4 Division Properties of Exponents
•
Use division properties of exponents to evaluate and simplify
expressions.
✔ SAT-10 ✔ NAEP ✔ ACT
✔ SAT
✔ SAT Subject Tests
□
□
□
□
Technology Lab Activities
7-4 Technology Lab
□
Lesson 7-5 Rational Exponents
• Evaluate and simplify expressions containing rational exponents.
✔ SAT-10 □
✔ NAEP □
✔ ACT
✔ SAT
□
□
□ SAT Subject Tests
Note: If NAEP is checked, the content is tested on either the Grade 8 or Grade 12 NAEP assessment.
460A
Chapter 7
Optional
graphing calculator,
number cubes
MK = Manipulatives Kit
Math Background
EXPONENTS
Lesson 7-1
and to see that 10,000,000 is a power of 10, namely
10 7. Thus, 93,000,000 = 9.3 × 10 7.
Up to this point, students have worked primarily with
linear equations and linear inequalities. In Chapter 7,
students move toward more complex ideas as they
begin to study polynomials. Before beginning this
study, students must first have an understanding of
exponents.
In general, every positive real number may be written
in scientific notation, a × 10 n, where 1 ≤ a < 10 and
n is an integer. The value of a is called the coefficient.
A number line helps to visualize scientific notation.
On the number line below, several powers of 10 are
graphed.
One common difficulty students have with exponents
is the use of zero. For example, students are often
puzzled by the fact that any nonzero number raised to
the zero power is 1. It makes sense to think of 2 4 as a
product where 2 is a factor 4 times (2 4 = 2 · 2 · 2 · 2),
but when it comes to evaluating 2 0, how does one
write a product with 2 as a factor zero times? Students
should understand that 2 0 is defined to be 1 in order
to make it consistent with the rules of exponent
arithmetic. For example, in order for the Quotient of
Powers Property to work in as many situations as
24
= 2 4-4 = 2 0,
possible, it must be true that __
24
4
2
but __4 = 1. Thus, 2 0 = 1.
The interval between each successive power of 10 is
10 times as large as the preceding interval. When
a number is written in scientific notation, such as
6.7 × 10 3, the power of 10 tells which interval the
number lies in and the coefficient tells where the
number falls within the interval.
101
102
103
10
1,000
100
6.7 × 103
104
2
This idea of defining certain powers in order to create
a system that is as widely consistent as possible also
explains why the expression 0 0 is undefined. First, it is
clear that 0 1 = 0, 0 2 = 0, and 0 13 = 0. In fact, for any
value of n greater than zero, 0 n = 0. For this reason, it
might make sense to define 0 0 as zero. On the other
hand, as shown above, any nonzero number raised to
the zero power is 1, so it might also make sense to
define 0 0 as 1. Because there is no single real number that works consistently as a definition of 0 0, this
expression is considered indeterminate and is
left undefined.
SCIENTIFIC NOTATION
LESSON 7-2
10,000
Numbers greater than or equal to 1 but less than 10
are written in scientific notation with an exponent of 0
since, for example,
3.8 = 3.8 × 1
= 3.8 × 10 0
Numbers greater than 0 but less than 1 are written
with negative powers of 10. A specific example shows
why this is the case:
0.0041 = 4.1 × 0.001
1
= 4.1 ×
1000
= 4.1 × 10 -3
_
Scientific notation is an efficient way to write very large
and very small numbers, such as numbers used to
express distances in space. For example, the distance
from the Earth to the Sun is approximately 93 million
miles or 93,000,000 miles.
The key step in the translation to scientific notation is
to recognize that
93,000,000 = 9.3 × 10,000,000
460B
7-1
Organizer
7-1
Block
Integer Exponents
__1 day
2
Objectives: Evaluate expressions
containing zero and integer
exponents.
GI
Simplify expressions containing
zero and integer exponents.
<D
@<I
Objectives
Evaluate expressions
containing zero and
integer exponents.
Who uses this?
Manufacturers can use negative exponents
to express very small measurements.
Simplify expressions
containing zero and
integer exponents.
In 1930, the Model A Ford was one of the first
cars to boast precise craftsmanship in mass
production. The car’s pistons had a diameter
of 3 _78_ inches; this measurement could vary
by at most 10 -3 inch.
Online Edition
Tutorial Videos
Countdown Week 15
You have seen positive exponents. Recall
that to simplify 3 2, use 3 as a factor 2 times:
3 2 = 3 · 3 = 9.
But what does it mean for an exponent to
be negative or 0? You can use a table and look for a pattern
to figure it out.
Warm Up
Base
x4
Evaluate each expression
for the given values of the
variables.
Exponent
Power
55
54
53
52
51
Value
3125
625
125
25
5
÷5
1. x3 y2 for x = -1 and y = 10
-100
3x2
2. _
for x = 4 and y = -7
y2
48
49
Write each number as a power
of the given base.
5 =1
50 = _
5
5 -2
÷5
1
1 =_
1 ÷5=_
5 -2 = _
5
25 5 2
1
1 =_
5 -1 = _
5 51
Integer Exponents
43
4. -27; base -3
÷5
5 -1
When the exponent decreases by one, the value of the power is divided by 5.
Continue the pattern of dividing by 5:
_
3. 64; base 4
÷5
50
WORDS
(-3)3
NUMBERS
Zero exponent—Any
nonzero number raised to
the zero power is 1.
Also available on transparency
Negative exponent—A
nonzero number raised to a
negative exponent is equal
to 1 divided by that number
raised to the opposite
(positive) exponent.
2 -4 is read “2 to
the negative fourth
power.”
no symbol for zero. It was represented by a blank space.
3 = 1 123 = 1
0
(-16) 0 = 1
0
ALGEBRA
If x ≠ 0, then x 0 = 1.
(_37 ) = 1
0
1 =_
1
3 -2 = _
9
32
1 =_
1
2 -4 = _
16
24
If x ≠ 0 and n is an integer,
1.
then x -n = _
xn
Notice the phrase “nonzero number” in the table above. This is because 0 0 and
0 raised to a negative power are both undefined. For example, if you use the
pattern given above the table with a base of 0 instead of 5, you would get 0 0 = __00 .
1
Also, 0 -6 would be __
= _10_. Since division by 0 is undefined, neither value exists.
6
0
460
Chapter 7 Exponents and Polynomials
1 Introduce
A1NL11S_c07_0460-0465.indd
e x p l o r460
at i o n
Motivate
7-1 Integer Exponents
A botanist has taken over a study of a plant whose height doubles
every day. On the fourth day after the experiment began, the plant
was 16 inches tall.
1. The botanist can find the height
of the plant on previous days by
repeatedly dividing by 2. Use
this fact to complete the middle
column of the table.
exponents to fill in the right
column for days 1 and 2.
Then look for patterns to
complete the right column.
3. What does day 0 represent?
What was the height of the plant
on day 0? How can you write the
height as a power?
KEYWORD: MA7 Resources
Day
Height
Height
of Plant Written as
(in.)
a Power
4
4
16
2
3
8
23
2
1
0
⫺1
⫺2
⫺3
4. What does day ⫺1 represent?
What was the height of the plant on day ⫺1? How can you
write the height as a power?
THINK AND DISCUSS
460
Chapter 7
5. Describe the pattern in the right column of the table.
6. Show how you could find the height of the plant on day ⫺4
and then write the height as a power.
Show students the following examples and ask
them to suggest a rule about the use of negative
exponents.
1
1
1
a. 2-1 = _
b. 2-3 = _
c. 5-2 = _
2
25
8
1
d. 3-2 = _
9
Possible answer: The negative exponent means
that you must use the reciprocal of the base
and change the exponent to a positive number.
Explorations and answers are provided in
Alternate Openers: Explorations Transparencies.
6/25/09 9:06:08 AM
EXAMPLE
1
Manufacturing Application
""
The diameter for the Model A Ford piston could vary by at most 10 -3 inch.
Simplify this expression.
Students will sometimes multiply the
base by the negative exponent. Have
of a negative exponent. Point out
that 10-3 means a number less than
one, not a number less than zero.
1 =_
1
1
10 -3 = _
=_
10 3 10 · 10 · 10 1000
1
inch, or 0.001 inch.
10 -3 inch is equal to ____
1000
A sand fly may have a wingspan up to 5 -3 m. Simplify this
1
expression.
1.
Ê,,",
,/
_m
125
EXAMPLE
2
Zero and Negative Exponents
Simplify.
A 2 -3
1 =_
1
1
2 -3 = _
=_
23 2 · 2 · 2 8
5 0 = 1 Any nonzero number raised
to the zero power is 1.
C (-3)-4
In (-3) -4, the base
is negative because
the negative sign
is inside the
parentheses.
In -3 -4 the base (3)
is positive.
Simplify.
1
B. 7 0 1
A. 4-3 _
64
1
1
C. (-5)-4 _ D. -5-4 - _
625
625
D -3 -4
1 = -_
1
1
-3 -4 = - _
= -_
3·3·3·3
81
34
Example 3
1
1
_
2b. (-2) _
-4
10,000
3
16
_
_
32
32
2c. (-2)-5- 1 2d. -2 -5 - 1
Evaluating Expressions with Zero and Negative Exponents
Evaluate each expression for the given value(s) of the variable(s).
A x -1 for x = 2
2 -1
2
-1
Substitute 2 for x.
1 =_
1
=_
21 2
1
1·_
1
__
Questioning Strategies
Write the power in the denominator as a product.
(-2)(-2)(-2)
_
_
INTERVENTION
Simplify expressions with exponents.
(-2)3
3a. 1
64
3b. 2
1· 1
-8
1
-_
8
3a. p
• What is the difference between
Examples 2C and 2D?
Simplify.
• When will a term with a negative
exponent have a negative value?
-2 0
for p = 4
3b. 8a b for a = -2 and b = 6
7-1 Integer Exponents
A1NL11S_c07_0460-0465.indd 461
3
• There are no fractions in the problem. Why are there fractions in the
Inclusion In Example 3,
some students may prefer
to rewrite the expression
with a positive exponent before substituting for the variable.
6/25/09 9:06:15 AM
Guided Instruction
Technology Students can use
the
or Yx keys on their
calculators to check their work
in Examples 1—3.
461
EX AM P LE
• Why is the value of a variable irrelevant if that variable is raised to
the zero power?
2 Teach
Explain negative and zero exponents by
demonstrating the pattern in the decreasing values of the powers. Use the definition
to simplify and evaluate expressions with
negative and zero exponents. Then include
examples with negative exponents in the
denominator. Remind students that factors
with negative exponents should be
simplified.
EX AM P LES 1 – 2
Simplify the power in the denominator.
Evaluate each expression for the given value(s) of the variable(s).
-3
Evaluate each expression
for the given value(s) of the
variable(s).
1
A. x -2 for x = 4 _
16
B. -2a 0b-4 for a = 5 and
2
b = -3 - _
81
1 .
Use the definition x -n = _
xn
B a 0b -3 for a = 8 and b = -2
8 0 · (-2)-3
Substitute 8 for a and -2 for b.
1·
One cup is 2-4 gallons. Simplify
1
this expression. _ gal
16
Example 2
1 = __
1
1
(-3)-4 = _
=_
(-3) 4 (-3)(-3)(-3)(-3) 81
Simplify.
2a. 10 -4
EXAMPLE
Example 1
B 50
Through Graphic Organizers
Have students make a chart similar to the
one below and let them refer to it during
class work and homework.
0 x (x ≤ 0)
Undefined
x 0 (x ≠ 0)
1
x -2 (x ≠ 0)
1
_
x2
1
_
(x ≠ 0)
x -2
x2
Lesson 7-1
461
What if you have an expression with a negative exponent in a denominator,
1
?
such as ___
-8
x
x
Example 4
Simplify.
7
A. 7w -4 _4
w
a0b -2
C. _
c -3d 6
-5
B. _
k-2
-n
1 = x -n
1 , or _
=_
xn
xn
1 = x -(-8)
_
x -8
Definition of negative exponent
Substitute -8 for n.
= x8
-5k2
Simplify the exponent on the right side.
So if a base with a negative exponent is in a denominator, it is equivalent to the
same base with the opposite (positive) exponent in the numerator.
c3
_
2
b d6
An expression that contains negative or zero exponents is not considered to be
simplified. Expressions should be rewritten with only positive exponents.
EXAMPLE
INTERVENTION
4
Simplifying Expressions with Zero and Negative Exponents
Simplify.
Questioning Strategies
-4
B _
-4
A 3y -2
EX A M P L E
4
k
-4 = -4 · 1
_
k -4
k -4
_
3y -2 = 3 · y -2
• How do you decide which factors
get moved to the other side of the
fraction bar?
_
= 3 · 12
y
3
=_
y2
• What happens to factors with exponents of zero?
= -4 · k 4
= -4k 4
x -3
C _
0 5
a y
x -3 = _
1
_
a 0y 5 x 3 · 1 · y 5
1
=_
x 3y 5
Simplify.
4a. 2r 0m -3
1.
a 0 = 1 and x -3 = _
x3
2
_
m
3
1
r -3 _
4b. _
7 7r 3
g4
4c. _
g 4h 6
h -6
THINK AND DISCUSS
-3
s =_
2,_
1 , ? -2 = _
1
1. Complete each equation: 2b ? = _
b2 k ?
s3
t2
2. GET ORGANIZED Copy and complete the graphic organizer. In each
box, describe how to simplify, and give an example.
-«vÞ}ÊÝ«ÀiÃÃÃÊÜÌÊ
i}>ÌÛiÊÝ«iÌÃ
ÀÊ>Êi}>ÌÛiÊiÝ«iÌÊ
ÊÌiÊÕiÀ>ÌÀÊ°Ê°Ê°
462
Chapter 7 Exponents and Polynomials
3 Close
Summarize
1. -2; 0; t
A1NL11S_c07_0460-0465.indd 462
and INTERVENTION
Remind students that an expression is
not simplified if it has an exponent that is
negative or zero.
Diagnose Before the Lesson
7-1 Warm Up, TE p. 460
Have students state the rules for simplifying expressions with negative exponents
in their own words. Accept nontechnical
answers such as the following: Make the
exponent positive and move the factor to
the other side of the fraction bar.
Monitor During the Lesson
Check It Out! Exercises, SE pp. 461–462
Questioning Strategies, TE pp. 461–462
462
Chapter 7
ÀÊ>Êi}>ÌÛiÊiÝ«iÌÊ
ÊÌiÊ`i>ÌÀÊ°Ê°Ê°
Assess After the Lesson
7-1 Lesson Quiz, TE p. 465
Alternative Assessment, TE p. 465
2. See p. A6.
7/18/09 4:51:41 PM
7-1
Exercises
7-1 Exercises
KEYWORD: MA7 7-1
KEYWORD: MA7 Parent
GUIDED PRACTICE
SEE EXAMPLE
1
p. 461
Assignment Guide
1. Medicine A typical virus is about 10 -7 m in size. Simplify this expression.
1
_
m
10,000,000
SEE EXAMPLE
2
Simplify.
1
_
3. 3 1
36
1
1 8. 10 _
7. -8 - _
2. 6 -2
p. 461
-2
100
512
SEE EXAMPLE
3
p. 461
_
_
_
12. b -2 for b = -3 1
13. (2t)-4 for t = 2
_
1
14. (m - 4)-5 for m = 6
32
p. 462
If you finished Examples 1–2
Basic 24—36, 77, 86, 88
Average 24—36, 77, 86, 88
Evaluate each expression for the given value(s) of the variable(s).
9
SEE EXAMPLE 4
_
_
1
- 1
-3
6. 1 -8 1
25 5. 3 27
0
-3
9. (4.2) 1
10. (-3) - 1 11. 4-2 1
16
27
4. -5 -2
0
-3
Assign Guided Practice exercises
as necessary.
20. 2x 0y -4
17. 3k -4
y
_3
k4
_
_2
6
f -4 g
21. _
g -6 f 4
4
256
15. 2x 0y -3 for x = 7 and y = -4 -
Simplify.
16. 4m 0 4
1
_
If you finished Examples 1–4
Basic 24—77, 86—99,
102—113
Average 24—57, 58—74 even,
76—100, 102—113
77—113
1
_
32
7
18. _
7r 7
r -7
x 10
19. _
x 10d 3
d -3
c4
22. _
c 4d 3
d -3
23. p 7q -1
p
_
7
q
Homework Quick Check
Quickly check key concepts.
Exercises: 24, 28, 34, 42, 52, 77
PRACTICE AND PROBLEM SOLVING
Independent Practice
For
See
Exercises Example
24
25–36
37–42
43–57
1
2
3
4
Extra Practice
Skills Practice p. S16
Application Practice p. S34
24. Biology One of the smallest bats is the northern blossom bat,
which is found from Southeast Asia to Australia. This bat weighs
about 2 -1 ounce. Simplify this expression. 1 oz
_
2
Simplify.
26. 5 -4
25. 8 0 1
1
_
625
1
_
_
49
36
1 34. (-4) 16
33. (-3) - _
29. -6 -2
- 1 30. 7 -2
-1
2
3
1
_
27. 3 -4
81
2 01
31. _
(5)
1
35. (_
2)
-2
4
1
_
81
1
_
169
1
-_
28. -9 -2 32. 13 -2
36. -7 -1
7
Evaluate each expression for the given value(s) of the variable(s).
37. x -4 for x = 4
1
_
38.
256
39. (10 - d) for d = 11 1
0
43. k -4
_1
k4
44. 2z -8
x 6y 2
53. b 0c 0 1
for v = 9
40. 10m n
1
_
_2
z8
0
_
7
1
_
216
for m = 10 and n = -2 -
32
_
_s
r5
_
46. c -2d
_d
c
2
s 5 s 5t 12
51. _
t -12
47. -5x -3 -
_5
x3
6
3w -5 3x
52. _
-6
5
x
w
_
_
3 -1
q -2r 0 1
h3
a -7b 2
k
2 m -1n 5 55. _
_
54. _
56. _
57. h
5
2
4
0
3
-4
2
2n
3
s
c d b d
6m
6m 2k
q2
_
3m
_
a 7c 3
7-1 Integer Exponents
A1NL11S_c07_0460-0465.indd 463
1
_
42. 4w vx v for w = 3, v = 0, and x = -5 4
3
1 b
45. _
2b -3 2
2f
r -5
_
48. 4x -6y -2 4 49. _
2g 10 50. s -1
7g -10
_
-3
-1 -5
-2
1 and b = 8
41. (3ab) for a = _
2
144
Simplify.
(_23 v)
463
6/25/09 9:06:34 AM
KEYWORD: MA7 Resources
Lesson 7-1
463
Evaluate each expression for x = 3, y = -1, and z = 2.
Exercise 94 involves
writing an equation with a negative
exponent. This exercise prepares
students for the Multi-Step Test Prep
on page 494.
_1
58. z -5
62.
66.
59.
1
_
1
77. red blood cell: ______
m; white
125,000
3
m; platelet:
blood cell: ______
250,000
Simplify.
61. (xyz)-1 -
_1
1
6
.q
a
_
68. c
3
67. a 3b -2
Biology
,
*
XXXXXXX
,
.
.q , XXXXX
q,
.q
d
_
3
-4
d3
_
_
_
4
_
_
77. Biology Human blood contains red blood
cells, white blood cells, and platelets. The table
shows the sizes of these components. Simplify
each expression.
23 = 8, 22 = 4,
21 = 2, 20 = 1,
1
1
2-1 = _, 2-2 = _,
4
2
1
1
2-3 = _ = _
8
23
60. (yz) 0 1
16
0
5
2
69. v 0w 2y -1 w 70. (a 2b -7) 1 71. -5y -6 c
b2
y6 3
y
-5
3
2
-8
20p -1
4q
6
2a
2a
m
x
3
_
_
_
_
_
1 76. 2b 73. -1 2a b 74. -3
72.
75.
p
b -6
b
n
3y 12
5q -3
m 2n 3
8 12
5
3x y
a
3
________
m
1,000,000
85. Possible answer: Look at the
pattern below. As the exponent
goes down by 1, the value is half
of what it was before.
1
_
Explain the error.
66. Equation A is incorrect because
5 was incorrectly moved to the
denom. The neg. exp. applies only
to the base x.
(x + y)-4
63. x -y 3
64. (yz) -x - _
65. xy -4 3
(xy - 3)-2 36
8
/////ERROR ANALYSIS///// Look at the two equations below. Which is incorrect?
32
Blood Components
Part
125,000 -1
Red blood cell
Tell whether each statement is sometimes,
always, or never true.
When bleeding occurs,
platelets (which
appear green in the
image above) help to
form a clot to reduce
blood loss. Calcium
and vitamin K are
also necessary for clot
formation.
Size (m)
White blood cell
3(500)-2
Platelet
3(1000)-2
1
78. If n is a positive integer, then x -n = __
. always
xn
79. If x is positive, then x -n is negative. never
80. If n is zero, then x -n is 1. sometimes
81. If n is a negative integer, then x -n = 1. sometimes
82. If x is zero, then x -n is 1. never
83. If n is an integer, then x -n > 1. sometimes
84. Critical Thinking Find the value of 2 3 · 2 -3. Then find the value of 3 2 · 3 -2. Make a
conjecture about the value of a n · a -n. 1; 1; a n · a -n = 1
1
85. Write About It Explain in your own words why 2 -3 is the same as __
.
3
2
Find the missing value.
1 =2
86. _
4
-2
1
90. 7 -2 = _
49
1
87. 9 -2 = _
81
91. 10
1 =
88. _
64
-2
89. _ = 3 -1 1
3
1 =2·5
93. 2 · _
-1
5
8
3
1
=_
-3 92. 3 · 4 -2 = _
16
1000
94. This problem will prepare you for the Multi-Step Test Prep on page 494.
a. The product of the frequency f and the wavelength w of light in air is a constant v.
Write an equation for this relationship. f w = v
b. w = v ; w = v f -1
f
b. Solve this equation for wavelength. Then write this equation as an equation with
f raised to a negative exponent.
c. The units for frequency are hertz (Hz). One hertz is one cycle per second, which
-1
is often written as __1s . Rewrite this expression using a negative exponent. s
_
7-1 PRACTICE A
7-1 PRACTICE C
_______________________________________
___________________
__________________
Practice B
7-1 PRACTICE B
Integer Exponents
LESSON
7-1
464
Chapter 7 Exponents and Polynomials
Simplify.
1. 53 =
3. (5)2
5. 60
1
53
1
25
=
1
2. 26 =
125
4. (4)3
1
6. (7)2
1
26
Name _______________________________________ Date___________________ Class __________________
1
=
64
8. a5b6 for a = 3 and b = 2
1
8
1
64
1
49
10. 5zx for z = 3 and x = 2
11. (5z)x for z = 3 and x = 2
5
9
1
9
A1NL11S_c07_0460-0465.indd
12. c3 (162) for c = 4
1
225
1
16,384
Simplify.
13. t4
14. 3r5
s 3
t 5
5
1
3
t
t4
r5
s3
2x 3 y 2
17.
z4
h0
16.
3
1
3
19.
15.
4fh
20.
7c
a 4 c 2e 0
b 1d 3
a4 bc2 d 3
10a 4 b
21.
k 5 =
1
k5
2. What number can go in the box to make a true statement: 5
1
3. Write the expression 3 with a negative exponent.
8
3g 2 hk 2
6h0
Chapter 7
1
=1
2
53 =
0
31
24 =
5. 25
32
100,000
7. 70
1
inch or 0.125 inch
8
9. ( 4)3
8.
13. 8x
64
1
5
10.
1
106
1,000,000
1
( 4)3 64
c2
2 3
12. c d
t4
8
x5
8
b
1
32
6. 25
1
-3
1
Simplify each expression.
h
2g 2 k 2
14. 12r
d3
0
12
1
= 82
82
1
24
1
= 24
24
Simplify 4 2.
Simplify x2y 3z0.
x2y3z0
Write without negative exponents.
x 2 z0
y3
Write in expanded form.
x 2 (1)
y3
z0 = 1.
Simplify.
x2
y3
Simplify.
Write without negative
exponents.
6
0
= 1?
4. What is the reciprocal of b7?
Negative Exponents
in the Denominator
42
1
42
1. What is the base of the expression 6 ?
3
11. t 4
464
0
60 = 1
Negative Exponents
For any nonzero number x For any nonzero number
1
x and any integer n,
and any integer n, xn = n .
1
x
= xn.
x n
For any nonzero
number x,
x0 = 1.
00 and 0n are undefined.
4
23. A ball bearing has diameter 23 inches.
Evaluate this expression.
Examples
1
= m3
m 3
22.A cooking website claims to contain 105 recipes.
Evaluate this expression.
Zero Exponents
Definition
5g 5
x 3 y 2z 4
14a 4
20bc 1
7-1 RETEACH
Integer Exponents
Positive exponents: The answer is the
base multiplied by itself the number of
times identified by the exponent.
Zero exponent: The answer is always
1 (if the base is not 0; 00 is undefined).
1
1
3 –2 =
=
3•3 9
1
1
3 –3 =
=
the reciprocal of the same expression
3 • 3 • 3 27 with a positive exponent.
1
1
3 –4 =
= 3 • 3 • 3 • 3 81
Note that the rules are the same when the base is a variable:
g0 = 1
Review for Mastery
7-1
Remember that 23 means 2 2 2 = 8. The base is 2, the exponent is positive 3.
Exponents can also be 0 or negative.
30 = 1
1
3 –1 =
3
b3 = b • b • b
4fg 5
18.
5h 3
2
34 = 3 • 3 • 3 • 3 = 81 33 = 3 • 3 • 3 = 27
32 = 3 • 3 = 9
46431 = 3
Name _______________________________________ Date___________________ Class __________________
LESSON
Studying the patterns that are found in expressions with exponents can
exponents.
9. (b 4)2 for b = 1
243
64
Using Patterns
7-1
Evaluate each expression for the given value(s) of the variable(s).
7. d3 for d = 2
LESSON
7
1
4•4
1
16
Fill in the blanks to simplify each expression.
1. 25
2. 103
25 =
1
3.
103 =
5
2
1
1
=
25
2•2•2•2•2
1
1
=
103
10 • 10 • 10
1
=
32
1
54
1
4
= 5
54
1
10
5
4
1
=
1000
= 5•5•5•5
= 625
Simplify.
4. 5y 4
5
y4
5.
x3
7. 1
x y
x4
y
b2
8. 1 3
a b
8
a 3
8a 3
a
b
6. 9x 3 y 2
4
9. 5x y
2
9x 3
y2
5y 2
x4
6/25/09 9:06:41 AM
In Exercise 96, if
students chose F, they
may have used the
negative exponent as a factor. If they
chose G, they may have made the
base negative because the exponent
is negative.
95. Which is NOT equivalent to the other three?
1
_
25
5 -2
0.04
-25
(-6)(-6)
1
-_
6·6
1
_
6·6
a 3b 2
_
-c
a3
_
-b 2c
c
_
a 3b 2
96. Which is equal to 6 -2?
6 (-2)
If students chose D in Exercise 97,
they probably moved every factor to
the opposite side of the fraction bar,
even if it had a positive exponent.
3 -2
ab .
97. Simplify _
c -1
a 3c
_
2
b
_5 , or 1.25
98. Gridded Response Simplify ⎡⎣2 -2 + (6 + 2)0⎤⎦.
4
99. Short Response If a and b are real numbers and n is a positive integer, write a
simplified expression for the product a -n · b 0 that contains only positive exponents.
1 ; a -n = 1 and b 0 = 1 for b ≠ 0.
_
CHALLENGE AND EXTEND
100.
rapidly as x increases.
-4
1
16
x
__
y = 2x
x
_
100. Multi-Step Copy and complete the table of values below. Then graph the ordered
pairs and describe the shape of the graph. Possible answer: y increases more
y
_
_
an
an
So you have 1n · 1, or simply 1n .
a
a
-3
1
8
-2
1
4
__
-1
1
2
__
__
Journal
0
1
2
3
4
1
2
4
8
16
101. Multi-Step Copy and complete the table. Then write a rule for the values of 1 n and
n
n
(-1)n when n is any negative integer. 1 n = 1; (-1) = -1 if n is odd, and (-1) =
Have students use patterns to
explain why any number raised to
the zero power, except zero, is one.
1 if n is even.
-1
-2
1n
1
1
1
1
1
(-1)n
-1
1
-1
1
-1
n
-3
-4
-5
Have students choose three exercises from Exercises 37—42, write
each expression in words, and then
show two different ways to evaluate
each expression.
SPIRAL REVIEW
Solve each equation. (Lesson 2-3)
y
104. _ - 8 = -12 -20
5
102. 6x - 4 = 8 2
103. -9 = 3 (p - 1) -2
105. 1.5h - 5 = 1 4
1 n + 2 - n 28
106. 2w + 6 - 3w = -10 16 107. -12 = _
2
Identify the independent and dependent variables. Write a rule in function notation
for each situation. (Lesson 4-3)
110. y = 3x - 4
111. y = 1 x + 5
3
112. y = 2
3
113. y = -4x + 9
_
_
108. Pink roses cost \$1.50 per stem. ind.: number of roses; dep.: total cost; f (x) = 1.50x
109. For dog-sitting, Beth charges a \$30 flat fee plus \$10 a day.
ind.: number of days; dep.: total cost; f (x) = 10x + 30
Write the equation that describes each line in slope-intercept form. (Lesson 5-7)
110. slope = 3, y-intercept = -4
111. slope = _13_, y-intercept = 5
112. slope = 0, y-intercept = __23
113. slope = -4, the point (1, 5) is on the line.
7-1 Integer Exponents
________________________________________
LESSON
7-1
___________________
__________________
Problem Solving
7-1 PROBLEM SOLVING
Integer Exponents
1. At the 2005 World Exposition in Aichi,
Japan, tiny mu-chips were embedded in
counterfeiting. The mu-chip was
developed by Hitachi in 2003. Its area
42(10)−2 square millimeters. Simplify
A1NL11S_c07_0460-0465.indd is465
this expression.
2. Despite their name, Northern Yellow
Bats are commonly found in warm,
humid areas in the southeast United
States. An adult has a wingspan of
about 14 inches and weighs between
3(2)−3 and 3(2)−2 ounces. Simplify these
expressions.
3
3
and
oz
4
8
4
or 0.16 mm 2
25
3. Saira is using the formula for the area
of a circle to determine the value of π.
She is using the expression Ar−2 where
A = 50.265 and r = 4. Use a calculator
to evaluate Saira’s expression to find her
approximation of the value of π to the
nearest thousandth.
4. The volume of a freshwater tank can
be expressed in terms of x, y, and z.
Expressed in these terms, the volume
of the tank is x3y−2z liters. Determine
the volume of the tank if x = 4, y = 3,
and z = 6.
42
3.142
2
liters
3
Alison has an interest in entomology, the study of insects. Her
collection of insects from around the world includes the four
specimens shown in the table below. Select the best answer.
Insect
Mass
Emperor Scorpion
2−5 kg
African Goliath Beetle
11−1 kg
Giant Weta
2−4 kg
5−3 kg
A −
6. Many Giant Wetas are so heavy that
they cannot jump. Which expression is
another way to show the mass of the
specimen in Alison’s collection?
F −(2)4 kg
§ 1·
G ¨ ¸
H
1
kg
2•2•2•2
−4
kg
J 4
1
kg
2
1
kg
125
1
kg
125
C
1
kg
15
D 125 kg
7. Scorpions are closely related to spiders
and horseshoe crabs. What is the mass
of Alison’s Emperor Scorpion expressed
as a quotient?
1
kg
A −
32
B
1
kg
25
LESSON
7-1
C
1
kg
32
Challenge
___________________
__________________
7-1 CHALLENGE
Exploring Patterns in the Units Digit of xn
When you write out the first several powers of xn, where x and n are
positive integers, you can discover interesting patterns in the units digits of xn.
x1
x=2
21 = 2
1
x2
x3
x4
22 = 2(2) = 4
23 = 2(4) = 8
24 = 2(8) = 16
x5
5
2
x6
25 = 2(16) = 32 26 = 2(32) = 64
6
Notice that 2 and 2 have the same units digit and that 2 and 2 have
the same units digit. In the exercises that follow, you can discover other
number patterns involving the units digits of xn.
1.
x=1
2.
x=2
3.
x=3
4.
x=4
5.
x=5
6.
x=6
7.
x=7
x=8
9.
x=9
10.
x = 10
x1
x2
x3
x4
x5
x6
x7
x8
x9
1
2
3
4
5
6
7
8
9
0
1
4
9
6
5
6
9
4
1
0
1
8
7
4
5
6
3
2
9
0
1
6
1
6
5
6
1
6
1
0
1
2
3
4
5
6
7
8
9
0
1
4
9
6
5
1
8
7
4
5
6
3
2
9
0
1
6
1
6
5
6
1
6
1
0
1
2
3
4
5
6
7
8
9
0
9
4
1
0
465
1. A square foot is 3-2 square
yards. Simplify this
1
expression. _ yd2
9
Simplify.
1
2. 2 -6 _
64
1
3. (-7)-3 - _
343
5. -112 -121
4. 60 1
Evaluate each expression
for the given value(s) of the
variable(s).
1
6. x-4 for x = 10 _
10,000
7. 2a -1b -3 for a = 6 and
1
b=3 _
81
Simplify.
-3
4
9. _
-3y 6
8. 4y -5 _5
y
y -6
7/18/09 4:52:07 PM
In Exercises 1–10, find the first nine powers of each value of x. Using
the units digit of each result, complete the table. You may find a
calculator useful.
8.
5. Cockroaches have been found on every
continent, including Antarctica. What is
the mass of Alison’s Madagascar Hissing
Cockroach expressed as a quotient?
B
________________________________________
7-1
x -4
10. _
a 0y 3
1
_
x 4y 3
Also available on transparency
Refer to the table that you completed in Exercises 1–10. Describe the
pattern in the units digits of xn.
11. 1n For
all n, 1n has 1 as its units digit.
12. 2n The
pattern is 2, 4, 8, and 6, for n = 1, 2, 3, and 4 and then repeats.
13. 3n The
pattern is 3, 9, 7, and 1, for n = 1, 2, 3, and 4 and then repeats.
14. 5n For
all n > 0, 5 n has 5 as its units digit.
15. Write a rule that determines the units digit of 7n as a function of n.
If you divide n by 4, then the units digit is 7, 9, 3, or 1, depending
on whether the remainder is 1, 2, 3, or 0, respectively.
D 32 kg
Lesson 7-1
465
7-2
Organizer
7-2
Block
__1 day
Powers of 10 and
Scientific Notation
2
Objectives: Evaluate and
multiply by powers of 10.
@<I
Online Edition
Tutorial Videos, Interactivity
The table shows relationships between
several powers of 10.
Vocabulary
scientific notation
Countdown Week 15
÷ 10
10 3
10
Value
1000
100
2. 123 ÷ 1000
0.123
3. 0.003 × 100
0.3
4. 0.003 ÷ 100
0.00003
5. 104
7. 230
÷ 10
÷ 10
10 1
10 0
10 -1
10 -2
10 -3
10
1
1 = 0.1
_
10
1 = 0.01
_
100
1 = 0.001
_
1000
WORDS
NUMBERS
Positive Integer Exponent
0.0001
10 4 = 1 0, 0 0 0
If n is a positive integer, find the value
of 10 n by starting with 1 and moving the
decimal point n places to the right.
1
4 places
Negative Integer Exponent
Also available on transparency
If n is a positive integer, find the value of
10 -n by starting with 1 and moving the
decimal point n places to the left.
EXAMPLE
1
Q: How did the number written
in scientific notation feel after he
changed into standard form?
A: Powerless.
÷ 10
Powers of 10
10,000
6. 10-4
÷ 10
× 10
× 10
× 10
× 10
× 10
× 10
• Each time you divide by 10, the exponent decreases by 1 and the decimal
point moves one place to the left.
• Each time you multiply by 10, the exponent increases by 1 and the decimal
point moves one place to the right.
Evaluate each expression.
123,000
2
Power
Warm Up
1. 123 × 1000
÷ 10
Nucleus of a silicon atom
⎧
⎨
⎩
<D
Convert between
standard notation and
scientific notation.
1 = 0.0 0 0 0 0 1
10 -6 = _
10 6
⎧
⎨
⎩
GI
Convert between standard notation
and scientific notation.
Why learn this?
Powers of 10 can be used to read and
write very large and very small numbers,
such as the masses of atomic particles.
(See Exercise 44.)
Objectives
Evaluate and multiply by
powers of 10.
6 places
Evaluating Powers of 10
Find the value of each power of 10.
A 10 -3
zeros to the right or
left of a number in
order to move the
decimal point in that
direction.
466
B 10 2
C 10
0
move the decimal
point three places
to the left.
move the decimal
point two places
to the right.
move the decimal
point zero places.
0. 0 0 1
1 0 0
1
0.001
100
Chapter 7 Exponents and Polynomials
1 Introduce
A1NL11S_c07_0466-0471.indd
e x p l o r466
at i o n
7-2
Powers of 10 and
Scientific Notation
You will need a calculator for this Exploration.
1. You can use the exponent key,
, on
your calculator to evaluate powers of
10. Use your calculator as needed to
complete the table.
Power of 10
Value
10 5
10 6
10
7
10
8
10
9
2. Look for patterns in the table. How is the exponent in each
power of 10 related to the value of that power of 10?
3. What happens when you try to use your calculator to evaluate
larger powers of 10, such as 1015?
Motivate
Have students copy the following numbers:
0.0000000000095
2,700,000,000,000,000,000,000
Ask them why the numbers are difficult to copy
accurately. They have many zeros.
Say that numbers used in science and technology
often contain many zeros. Scientific notation was
developed to make these numbers easier to
work with.
THINK AND DISCUSS
KEYWORD: MA7 Resources
4. Explain how you could write the value of 1015. How many
zeros would you write?
5. Describe a general rule you can use to write the value of
10 n, where n is a positive integer.
466
Chapter 7
Explorations and answers are provided in
Alternate Openers: Explorations Transparencies.
6/25/09 9:08:50 AM
Find the value of each power of 10.
1a. 10 -2 0.01
1b. 10 5 100,000
""
1c. 10 10
10,000,000,000
EXAMPLE
2
When writing the powers of 10 in
a chart or a list, students will often
start at 10, rather than 1. Remind
students that 100 = 1.
Writing Powers of 10
Write each number as a power of 10.
A 10,000,000
If you do not see
a decimal point
in a number, it is
understood to be
at the end of the
number.
B 0.001
C 10
The decimal point is
seven places to the
right of 1, so the
exponent is 7.
The decimal point
is three places to
the left of 1, so the
exponent is -3.
The decimal point
is one place to the
right of 1, so the
exponent is 1.
10 7
10 -3
10 1
Write each number as a power of 10.
2a. 100,000,000 10 8 2b. 0.0001 10 -4
Ê,,",
,/
Example 1
Find the value of each power
of 10.
2c. 0.1 10 -1
A. 10-6 0.000001
You can also move the decimal point to find the product of any number and a
Multiplying by Powers of 10
125 × 10 5 = 12,5 0 0, 0 0 0
If the exponent is a negative integer,
move the decimal point to the left.
36.2 × 10 -3 = 0.0 3 6 2
10,000
C. 109
1,000,000,000
Example 2
⎧
⎨
⎩
If the exponent is a positive integer,
move the decimal point to the right.
B. 104
Write each number as a power
of 10.
5 places
⎧
⎨
⎩
A. 1,000,000
3 places
10-4
B. 0.0001
EXAMPLE
3
103
Multiplying by Powers of 10
C. 1000
Find the value of each expression.
Example 3
A 97.86 × 10 6
97.8 6 0 0 0 0
Find the value of each
expression.
Move the decimal point 6 places to the right.
97,860,000
B 19.5 × 10
-4
0 0 1 9.5
106
Move the decimal point 4 places to the left.
A. 23.89 × 108
2,389,000,000
B. 467 × 10-3
0.467
0.00195
Find the value of each expression.
3a. 853.4 × 10 5 85,340,000
3b. 0.163 × 10 -2 0.00163
INTERVENTION
Scientific notation is a method of writing numbers that are very large or very
small. A number written in scientific notation has two parts that are multiplied.
The first part is a number that is greater than or equal to 1 and less than 10.
Questioning Strategies
EX AM P LE
1
• What does a positive exponent
represent?
• What does a negative exponent
represent?
The second part is a power of 10.
EX AM P LE
7- 2 Powers of 10 and Scientific Notation
2 Teach
6/25/09 9:08:59 AM
Guided Instruction
Show students the pattern of powers of 10.
First show some with positive exponents:
103
=
2
• What pattern do you notice when
you multiply repeatedly by 10?
• What pattern do you notice when
you divide repeatedly by 10?
A1NL11S_c07_0466-0471.indd 467
102 =
467
10 × 10
= 100
10 × 10 × 10
= 1000
Show how each is the number 1 followed
by the same number of zeros as the exponent number. Then work backward for 10
to the first power, zero power, and negative
powers. With negative exponents, the number of zeros is one less than the exponent
number.
EX AM P LE
Through Cooperative Learning
Separate students into groups of three.
The first student writes a number in standard form. The second student writes that
number in scientific notation, and the third
student checks/corrects the work. Have
students switch roles so that everyone has
done each job at least once. Then repeat,
having the first student write a number
in scientific notation, the second student
write that number in standard form, and
the third student check/correct the work.
3
• Why does multiplying by a negative power of 10 result in a smaller
number?
Lesson 7-2
467
EXAMPLE
4
Example 4
Saturn has a diameter of about
1.2 × 105 km. Its distance from
km.
from Earth in standard form.
5.91 × 10 8
5.9 1 0 0 0 0 0 0
Move the decimal point 8 places to the right.
Standard form
refers to the usual
way that numbers
are written.
B. Write Saturn’s distance from
the Sun in scientific notation.
1.427 × 109 km
591,000,000 km
B Write Jupiter’s average distance from the Sun in scientific notation.
⎩
⎨
⎧
778,400,000
7 7 8, 4 0 0, 0 0 0
Example 5
8 places
Order the list of numbers from
least to greatest.
×
×
×
×
£{Î]äääÊ
A Write Jupiter’s shortest distance
A. Write Saturn’s diameter in
standard form. 120,000 km
1.3
2.1
5.4
6.3
Astronomy Application
Jupiter has a diameter of about
143,000 km. Its shortest distance
from Earth is about 5.91 × 10 8 km,
and its average distance from the
Jupiter’s orbital speed is
approximately 1.3 × 10 4 m/s.
7.784 × 10 8 km
10-2, 6.3 × 103, 4.1 × 106,
106, 1 × 10-2, 5.4 × 10-3
10-3, 1 × 10-2, 1.3 × 10-2,
103, 2.1 × 106, 4.1 × 106
Count the number of places you need to move
the decimal point to get a number between 1
and 10.
Use that number as the exponent of 10.
4a. Use the information above to write Jupiter’s diameter in
scientific notation. 1.43 × 10 5 km
4b. Use the information above to write Jupiter’s orbital speed
in standard form. 13,000 m/s
EXAMPLE
5
Comparing and Ordering Numbers in Scientific Notation
Order the list of numbers from least to greatest.
1.2 × 10 -1, 8.2 × 10 4, 6.2 × 10 5, 2.4 × 10 5, 1 × 10 -1, 9.9 × 10 -4
INTERVENTION
Step 1 List the numbers in order by powers of 10.
Questioning Strategies
9.9 × 10 -4, 1.2 × 10 -1, 1 × 10 -1, 8.2 × 10 4, 6.2 × 10 5, 2.4 × 10 5
EX A M P L E
4
Step 2 Order the numbers that have the same power of 10.
• What does it mean to write a number in standard form?
EX A M P L E
9.9 × 10 -4, 1 × 10 -1, 1.2 × 10 -1, 8.2 × 10 4, 2.4 × 10 5, 6.2 × 10 5
5. Order the list of numbers from least to greatest.
5.2 × 10 -3, 3 × 10 14, 4 × 10 -3, 2 × 10 -12, 4.5 × 10 30, 4.5 × 10 14
5
2 × 10 -12, 4 × 10 -3, 5.2 × 10 -3, 3 × 10 14, 4.5 × 10 14, 4.5 × 10 30
• When ordering numbers in scientific notation, why is the power of 10
used to determine the initial order
of the numbers?
THINK AND DISCUSS
1. Tell why 34.56 × 10 4 is not correctly written in scientific notation.
Number Sense Tell
students to associate the
direction the decimal point
moves with the positive and negative directions on a number line:
positive numbers to the right and
negative numbers to the left.
2. GET ORGANIZED Copy and complete the graphic organizer.
*ÜiÀÃÊvÊ£äÊ>`Ê-ViÌvVÊ Ì>Ì
Êi}>ÌÛiÊiÝ«iÌ
VÀÀiÃ«`ÃÊÌÊÛ}ÊÌi
¶
`iV>Ê«ÌÊÚÚÚÚÚÚÚÚÚÚ°
468
Chapter 7 Exponents and Polynomials
3 Close
Summarize
1. 34.56 is not between 1 and 10.
A1NL11S_c07_0466-0471.indd 468
Remind students that to multiply by a positive power of 10, they should move the
decimal point to the right, and to multiply
by a negative power of 10, they should
move the decimal point to the left.
If a number is in scientific notation, it is
in the form x × 10 y, with 1 ≤ x < 10 and
with y being any integer.
468
Chapter 7
Ê«ÃÌÛiÊiÝ«iÌ
VÀÀiÃ«`ÃÊÌÊÛ}ÊÌi
¶
`iV>Ê«ÌÊÚÚÚÚÚÚÚÚÚÚ°
and INTERVENTION
Diagnose Before the Lesson
7-2 Warm Up, TE p. 466
Monitor During the Lesson
Check It Out! Exercises, SE pp. 467–468
Questioning Strategies, TE pp. 467–468
Assess After the Lesson
7-2 Lesson Quiz, TE p. 471
Alternative Assessment, TE p. 471
2. See p. A6.
6/25/09 9:09:03 AM
7-2
Exercises
7-2 Exercises
KEYWORD: MA7 7-2
KEYWORD: MA7 Parent
GUIDED PRACTICE
Assignment Guide
1. Vocabulary Explain how you can tell whether a number is written in scientific
notation. A number written in sci. notation is a product with 2 parts: a
decimal greater than or equal to 1 and less than 10 and a power of 10.
SEE EXAMPLE
1
SEE EXAMPLE
2
3
4. 10 -4 0.0001
SEE EXAMPLE 4
p. 468
5
p. 468
5. 10 8 100,000,000
Write each number as a power of 10.
7. 0.000001 10 -6
8. 100,000,000,000,000,000
10 17
Find the value of each expression.
9. 650.3 × 10 6 650,300,000 10. 48.3 × 10 -4 0.00483
p. 467
SEE EXAMPLE
3. 10 -5 0.00001
6. 10,000 10 4
p. 467
SEE EXAMPLE
Find the value of each power of 10.
2. 10 6 1,000,000
p. 466
11. 92 × 10 -3 0.092
12. Astronomy A light-year is the distance that light travels in a year and is equivalent
to 9.461 × 10 12 km. Write this distance in standard form. 9,461,000,000,000 km
13. Order the list of numbers from least to greatest.
8.5 × 10 -1, 3.6 × 10 8, 5.85 × 10 -3, 2.5 × 10 -1, 8.5 × 10 8
1
2
3
4
5
Extra Practice
Skills Practice p. S16
Application Practice p. S34
Find the value of each power of 10.
14. 10 3 1000
15. 10 -9
16. 10 -12
17. 10 14
0.000000001
0.000000000001
100,000,000,000,000
Write each number as a power of 10.
18. 0.01 10 -2
20. 0.000000000000001 10 -15
19. 1,000,000 10 6
Find the value of each expression.
21. 9.2 × 10 4 92,000 22. 1.25 × 10 -7
0.000000125
23. 42 × 10 -5
0.00042
24. 0.05 × 10 7
500,000
25. Biology The human body is made of about 1 × 10 13 cells. Write this number in
standard form. 10,000,000,000,000
27. Order the list of numbers from least to greatest.
2.13 × 10 -1, 3.12 × 10 2, 1.23 × 10 -3, 2.13 × 10 1, 1.32 × 10 -3, 3.12 × 10 -3
28.
28. Yes; the
smallest grain of
pollen is larger than
3 × 10 -7 m.
1.23 × 10 -3, 1.32 × 10 -3, 3.12 × 10 -3, 2.13 × 10 -1, 2.13 × 10 1, 3.12 × 10 2
Health Donnell is allergic to pollen. The diameter of a grain
of pollen is between 1.2 × 10 -5 m and 9 × 10 -5 m. Donnell’s
air conditioner has a filter that removes particles larger than
3 × 10 -7 m. Will the filter remove pollen? Explain.
29. Entertainment In the United States, a CD is certified platinum
if it sells 1,000,000 copies. A CD that has gone 2 times platinum
has sold 2,000,000 copies. How many copies has a CD sold if it has
2.7 × 10 7
Write each number in scientific notation.
to indicate the power of
10. For example, to enter 9.2 × 104,
press 9.2
4. To enter
or
powers of 10, use
-5
. To enter 10 , press 10
-5 or
(-5).
Grain of pollen,
enlarged 1300 times
1.7 × 10 11
30. 40,080,000 4.008 × 10 7 31. 235,000 2.35 × 10 5
32. 170,000,000,000
33. 0.0000006 6 × 10 -7
35. 0.0412 4.12 × 10 -2
34. 0.000077 7.7 × 10 -5
Number Sense In
Exercises 14—17, have
students estimate whether
their answer is greater than or less
than 10 before finding the value.
This will help them if they forget the
rules for evaluating powers of 10.
Technology For Exercises
21—24, students can enter
numbers in scientific notation into their calculators by using
26. Statistics At the beginning of the twenty-first century, the population of China
was about 1,287,000,000. Write this number in scientific notation. 1.287 × 10 9
7- 2 Powers of 10 and Scientific Notation
A1NL11S_c07_0466-0471.indd 469
If you finished Examples 1–5
Basic 14—46, 48—52,
55—63
Average 14—53, 55—63
Quickly check key concepts.
Exercises: 16, 18, 22, 26, 27, 36
PRACTICE AND PROBLEM SOLVING
14–17
18–20
21–24
25–26
27
If you finished Examples 1–3
Basic 14—24
Average 14—24, 53
Homework Quick Check
5.85 × 10 -3, 2.5 × 10 -1, 8.5 × 10 -1, 3.6 × 10 8, 8.5 × 10 8
Independent Practice
For
See
Exercises Example
Assign Guided Practice exercises
as necessary.
469
7/18/09 4:50:48 PM
KEYWORD: MA7 Resources
Lesson 7-2
469
State whether each number is written in scientific notation. If not, write it in
scientific notation.
Chemistry
40. 0.1
41. 7 × 10 8
42. 48,000
43. 3.5 × 10 -6
45. Communication This bar graph
shows the increase of cellular telephone
subscribers worldwide.
a. Write the number of subscribers for the
following years in standard form: 1999,
2000, and 2003.
40. no; 1 ×
45a. 490,000,000;
740,000,000;
1,329,000,000
41. yes
42. no; 4.8 ×
104
43. yes
49b. Possible answer: It would be
easy to accidentally omit a 0 or
add an extra 0 when writing the
number in standard form. You
are probably less likely to make
an error when using scientific
notation.
b. When you double
7.4 × 10 8, you get
approx. 14 × 10 8,
or 1.4 × 10 9 in
sci. notation. 1.4
is close to 1.3, so
Zorah’s observation
is correct.
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{ÊÊÊ£äÊnÊ
ÓÊÊÊ£äÊnÊ
£°ÎÓÊÊÊ£äÊ
nÊÊÊ£äÊnÊ
£°£xxÊÊÊ£äÊ
£°{ÊÊÊ£äÊÊ
£°ÓÊÊÊ£äÊÊ
£ÊÊÊ£äÊÊ
°xÊÊÊ£äÊn
b. Zorah looks at the bar graph and says,
“It looks like the number of cell phone
subscribers nearly doubled from 2000
to 2003.” Do you agree with Zorah?
Use scientific notation to explain
39. no; 2.5 × 102
10-1
7À`Ü`iÊ
iÊ*iÊ-ÕLÃVÀLiÀÃ
Ç°{ÊÊÊ£äÊn
38. no; 1.2 × 106
39. 0.25 × 10 3
{°ÊÊÊ£äÊn
37. yes
38. 1,200,000
electron
The image above is a
colored bubble-chamber
photograph. It shows
the tracks left by
subatomic particles in a
particle accelerator.
36. no; 5 × 10-4
37. 8.1× 10 -2
44. Chemistry Atoms are made of three elementary particles: protons, electrons,
and neutrons. The mass of a proton is about 1.67 × 10 -27 kg. The mass of an
electron is about 0.000000000000000000000000000000911 kg. The mass
of a neutron is about 1.68 × 10 -27 kg. Which particle has the least mass?
(Hint: Compare the numbers after they are written in scientific notation.)
Exercise 44, in
1.67 × 10 -27, 1.67 is called
the coefficient.
36. 50 × 10 -5
-ÕLÃVÀLiÀÃ
Exercise 49 involves
writing numbers in
scientific notation.
This exercise prepares students for
the Multi-Step Test Prep on
page 494.
£ Óäää Óää£ ÓääÓ ÓääÎ
9i>À
46. Measurement In the metric system, the basic unit for measuring length is the
meter (m). Other units for measuring length are based on the meter and powers of
10, as shown in the table.
b. 10 -3 = 0.001;
10 -2 = 0.01;
10 -1 = 0.1;
10 1 = 10;
10 2 = 100;
10 3 = 1000
Selected Metric Lengths
1 millimeter (mm) = 10 -3 m
1 centimeter (cm) = 10
1 decimeter (dm) = 10
-2
-1
1 dekameter (dam) = 10 1 m
m
1 hectometer (hm) = 10 2 m
m
1 kilometer (km) = 10 3 m
a. Which lengths in the table are longer than a meter? Which are shorter than a
meter? How do you know? dam, hm, km; mm, cm, dm
b. Evaluate each power of 10 in the table to check your answers to part a.
1 = 10 -3. Based on this information,
47. Critical Thinking Recall that ___
10 3
complete the following statement: Dividing a number by 10 3 is equivalent to
multiplying by . 10 -3
48. Write About It When you change a number from scientific notation to standard
form, explain how you know which way to move the decimal point and how many
places to move it. If the exp. is pos., move the dec. pt. that many places to the
right. If the exp. is neg., move the dec. pt. that many places to the left.
49. This problem will prepare you for the Multi-Step Test Prep on page 494.
8
a. The speed of light is approximately 3 × 10 m/s. Write this number in standard
form. 300,000,000
b. Why do you think it would be better to express this number in scientific
notation rather than standard form?
c. The wavelength of a shade of red light is 0.00000068 meters. Write this number
in scientific notation. 6.8 × 10 -7
7-2 PRACTICE A
7-2 PRACTICE C
________________________________________
LESSON
7-2
__________________
__________________
Practice B
7-2 PRACTICE B
Powers of 10 and Scientific Notation
470
Find the value of each power of 10.
1. 10−3
0.001
2. 105
100,000
3. 10−4
4. 100
1
5. 107
10,000,000
6. 101
LESSON
10
7-2
Write each number as a power of 10.
7. 1,000,000 10
10
10. 0.00001
6
−5
__________________
__________________
Use a Graphic Aid
The graphic aid below summarizes how to work with powers of 10.
8. 0.001 10
−3
10
−1
11. 0.1
Chapter 7 Exponents and Polynomials
________________________________________
0.0001
12. 0.00000001 10
5020
14. 603 × 10−4
0.0603
15. 52.8 × 106
52,800,000
16. 5.41 × 10−3
0.00541
17. 0.03 × 10−2
0.0003
18. 22.81 × 10−6
19. 4500
4.5 × 10
21. 0.00002
2 × 10−5
Find the value of 105.
Find the value of 10 −4.
1.0
Step 2: The exponent is positive 5. Move
the decimal 5 spaces to the right.
22. 0.00203
Step 2: The exponent is negative 4. Move
the decimal 4 spaces to the left.
1.0 0 0 0 0 = 100,000
0.00002281
0 . 0 0 1 = 0.0001
Write 100,000,000 as a power of 10.
20. 6,560,000
__________________
7-2 RETEACH
Powers of 10 and Scientific Notation
1.0
A1NL11S_c07_0466-0471.indd 470
Write each number in scientific notation.
3
__________________
Review for Mastery
The exponent will tell you how many places to move the decimal when
finding the value of a power of 10.
−8
Find the value of each expression.
13. 5.02 × 103
7-2
Powers of 10 are used to write large numbers in a simple way.
10−6
9. 0.000001
________________________________________
LESSON
Write 0.00001 as a power of 10.
1 0 0 0 0 0 0 0 0.
6.56 × 106
0.0 0 0 0 1
The decimal point is 8 places to the right of
the 1. The exponent is 8.
2.03 × 10−3
100,000,000 = 10
Order the list of numbers from least to greatest.
23. 3 × 102 ; 4.54 × 10−3 ; 6.75 × 102 ; 8.2 × 10−4 ; 9 × 10−1 ; 6.18 × 10−4
The decimal point is 5 places to the left of
the one. The exponent is −5.
0.00001 = 10−5
8
Numbers greater than 1 will
have a positive exponent.
6.18 × 10−4; 8.2 × 10−4; 4.54 × 10−3; 9 × 10−1; 3 × 102; 6.75 × 102
Numbers less than 1 will
have a negative exponent.
24. 5.4 × 10−3 ; 6.2 × 10−1 ; 7.25 × 103 ; 6.87 × 103 ; 2.24 × 10−1 ; 6.6 × 10−3
5.4 × 10−3; 6.6 × 10−3; 2.24 × 10−1; 6.2 × 10−1; 6.87 × 103;
7.25 × 103
25. In 1970, the number of televisions sold in the United States
was about 1.2 × 107. Write this number in standard form.
26. In 1950, about 3,880,000 households in the United States
had televisions. Write this number in scientific notation.
27. Find the volume of the cube shown at right. Write the
answer in both standard form and in scientific notation.
64,000,000,000 mm3
6.4 × 1010 mm3
First determine whether the decimal point will move to the right or to the left.
Then find the value of each power of 10.
Complete each of the following.
12,000,000
3.88 × 106
1. 106
1. Which represents a very large number: 4 × 109 or 4 × 10−9 ?
4 × 109
2. Write 9.7 × 10−3 in standard form.
0.0097
4.19 × 1011
3. Write 419,000,000,000 in scientific notation.
Match each number with its equivalent power of 10.
A. 104
B. 10 −5
C. 105
4. 0.00001 B
5. 10,000 A
6. 100,000 C
7. 0.0001 D
470
Chapter 7
56,000,000
9. 87.5 × 104
875,000
2. 10−2
3. 104
left
0.01
right
10,000
First determine whether the exponent will be positive or negative
when each number is written as a power of 10.
Then write each number as a power of 10.
4. 1000
D. 10 −4
Find the value of each expression.
8. 56 × 106
right
1,000,000
positive
103
5. 0.0001
negative
10−4
6. 10,000,000
positive
107
7/18/09 4:50:53 PM
If students chose A or
D in Exercise 50, they
may have associated
the exponent of 10 with the number
50. There are about 3.2 × 10 7 seconds in one year. What is this number in
standard form?
0.000000032
0.00000032
In Exercise 52, encourage students
to write the second and fourth numbers in the list in scientific notation
as a first step.
32,000,000
320,000,000
51. Which expression is the scientific notation for 82.35?
8.235 × 10 1
823.5 × 10 -1
8.235 × 10 -1
0.8235 × 10 2
52. Which statement is correct for the list of numbers below?
2.35 × 10 -8, 0.000000029, 1.82 × 10 8, 1,290,000,000, 1.05 × 10 9
The list is in increasing order.
If 0.000000029 is removed, the list will be in increasing order.
If 1,290,000,000 is removed, the list will be in increasing order.
The list is in decreasing order.
CHALLENGE AND EXTEND
53. About 7 times; 53. Technology The table shows estimates of
Computer Storage
computer storage. A CD-ROM holds 700 MB.
4.7 GB, the storage
1 kilobyte (KB) ≈ 1000 bytes
A DVD-ROM holds 4.7 GB. Estimate how many
of the DVD, is the
1 megabyte (MB) ≈ 1 million bytes
times
more
storage
a
DVD
has
than
a
CD.
same as 4700 MB,
1 gigabyte (GB) ≈ 1 billion bytes
which is approx. 7
times 700 MB, the
54. For parts a–d, use what you know about multiplying by powers of 10 and the
storage of the CD.
Commutative and Associative Properties of Multiplication to find each product.
Write each answer in scientific notation.
3
a. (3 × 10 2)(2 × 10 ) 6 × 10 5
b. (5 × 10 8)(1.5 × 10 -6) 7.5 × 10 2
c. (2.2 × 10 -8)(4 × 10 -3) 8.8 × 10 -11 d. (2.5 × 10 -12)(2 × 10 6) 5 × 10 -6
e. Based on your answers to parts a–d, write a rule for multiplying numbers in
scientific notation.
f. Does your rule work when you multiply (6 × 10 3)(8 × 10 5)? Explain.
54e. First multiply
the numbers, and
then multiply the
powers of 10 by adding the exponents.
Yes, but the answer, 48 × 10 8, is not in sci. notation. After multiplying, you
will have to rewrite the answer in sci. notation as 4.8 × 10 9.
SPIRAL REVIEW
Define a variable and write an inequality for each situation. Graph the solutions.
(Lesson 3-1)
Technology In
Exercise 53, you might
tell students that metric
prefixes are used to describe computer storage. However, bytes do not
example, 1 kilobyte = 1024 bytes,
not 1000 bytes.
55–57. For graphs, see p. A27.
Journal
Have students explain why 105 has 5
zeros, but 10-5 has only 4 zeros.
Have students write four numbers in
scientific notation, two with positive
exponents and two with negative
exponents, and then arrange them
from least to greatest.
55. Let m = number 55. Melanie must wait at least 45 minutes for the results of her test.
of minutes; m ≥ 45. 56. Ulee’s dog can lose no more than 8 pounds to stay within a healthy weight range.
7-2
56. Let p = pounds; 57. Charlene must spend more than \$50 to get the advertised discount.
p ≤ 8 where p is
nonneg.
Solve each system by elimination. (Lesson 6-3)
57. Let m = money 58. ⎧⎨ x + y = 8 (5, 3)
⎩x-y=2
spent; m > 50.
59.
⎧ 2x + y = -3 (-2, 1)
⎧ x - 6y = -3
60. ⎨
⎨
⎩ 2x + 3y = -1
⎩ 3x + 4y = 13
Find the value of each
expression.
(3, 1)
Evaluate each expression for the given value(s) of the variable(s). (Lesson 7-1)
_
61. t -4 for t = 2 1
16
62. (-8m)0 for m = -5 1
LESSON
7-2
7-2 PROBLEM SOLVING
Problem Solving
A1NL11S_c07_0466-0471.indd
1. Insects can multiply rapidly during the
summer. A pair of houseflies could
potentially grow to a population of
1.91 1020. If all the descendants of a
female cabbage aphid lived, the
population could increase to 1.56 1024.
471
Which population would be larger?
Name _______________________________________ Date __________________ Class__________________
LESSON
7-2
Powers of 10 and Scientific Notation
2. The graph shows the gross domestic
product (GDP) for several countries
around the world. Identify the country
whose GDP is twice that of another
country. Write the GDPs of both
countries in standard form.
3.38 107
Philippines
United Kingdom
8.79 10
6.04 107
1.
C
B 3.8 101 AU
D 38 10 2 AU
3. 8 10 1 AU
7. What is the diameter of the Earth in
scientific notation?
F 1.28 102 km
H
1.28 10 4 km
G 1.28 103 km J 1.28 105 km
a. How many times farther from the
sun is Pluto than Earth?
b. How many times farther from the
sun is Pluto than Jupiter?
5. Suppose the mass of Mars were written in
standard form. How many digits would be
to the left of the decimal?
F 23
G
B 1.50 109 km D 1.50 1011 km
A 0.38 AU
4.84 108 miles
is Jupiter than Earth?
1.50 108 km C 1.50 1010 km
6. Which of these is the average distance
from the Sun to Mercury expressed in
scientific notation?
9.3 107 miles
the sun in scientific notation.
List the countries in order of population
size from least to greatest.
A
a. Write the distance between Earth and
2. The average distance between Pluto and the sun
is 3,675,000,000 miles.
4. An AU is an astronomical unit. One AU
equals 150,000,000 km. What is that
measure in scientific notation?
H 25
24
3. The star closest to the sun is 25,000,000,000,000 miles
from the sun. How many times farther is it from the
sun to the nearest star as it is from Earth to the sun?
4. How many miles does light travel in a year? Give your
5.9 1012
answer in scientific notation rounded to the nearest tenth.
5.
a. One galaxy is 200,000 light-years from the sun.
J 26
b. How many times farther is this galaxy from
the sun than Earth is from the sun?
Planet
One light-year is the distance light travels in one year. Light
travels 186,282 miles in 1 second.
How many miles from the sun is the galaxy?
Astronomical Data for the First Five Planets
Avg. Distance Diameter Mass (kg)
from Sun (AU)
(km)
Mercury
0.38
4,880
3.20 1023
Venus
0.72
12,100
4.87 1024
Earth
1
12,800
5.97 1024
Mars
1.52
6,790
6.42 1023
Jupiter
5.20
143,000
1.90 1027
You can use scientific notation to compare masses of large
objects with masses of small objects.
Mass of hydrogen atom: 1.67 10 24 grams
Jerry’s mass:
Mass of Earth:
Holt McDougal Algebra 1
7/18/09 4:51:04 PM
a. Write the area of the Pacific
Ocean in standard form.
64,000,000 mi2
b. Write the volume of the
Pacific Ocean in scientific
notation. 1.7 × 108 mi3
4. Order the list of numbers from
least to greatest.
3.6 × 10 -3, 1 × 10 -5, 2.7 × 102,
1.3 × 104, 3.1 × 104, 4.1 × 10-3
1 × 10 -5, 3.6 × 10 -3, 4.1 ×
10 -3, 2.7 × 102, 1.3 × 104,
3.1 × 104
6. Write a ratio to compare Jerry’s mass with that of a hydrogen atom.
7-17
471
Also available on transparency
6.35 101 kilograms
5.97 1024 kilograms
7. Write a ratio to compare the mass of Earth with that of a hydrogen atom.
Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.
3. The Pacific Ocean has an area
of about 6.4 × 10 7 square
170,000,000 cubic miles.
7
Ethiopia: \$54,000,000,000;
Cambodia: \$27,000,000,000
125
How many times farther from the sun is Jupiter than Earth?
c. How many times farther from the Sun
Kenya, UK, Philippines,
Brazil, India
0.00293
7-2 CHALLENGE
and the sun in scientific notation.
The table shows astronomical data about several planets.
2. 29.3 × 10-4
Using Scientific Notation to Make Comparisons
b. Write the distance between Jupiter
3. The 2005 population estimates of five
countries are listed below.
Brazil
1.86 108
India
1.08 109
3
_
The average distance between Earth and the sun is 93,000,000 miles.
The average distance between Jupiter and the sun is 484,000,000 miles.
Using scientific notation, you can answer questions like the one below.
cabbage aphid
Kenya
Challenge
3,745,000
63. 3a -3b 0 for a = 5 and b = 6
7- 2 Powers of 10 and Scientific Notation
Name _______________________________________ Date __________________ Class__________________
1. 37.45 × 105
Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.
7-16
Holt McDougal Algebra 1
Lesson 7-2
471
7-3
Organizer
Explore Properties of
Exponents
Use with Lesson 7-3
Pacing:
You can use patterns to find some properties of exponents.
1
Block __
day
2
Objective: Use patterns to
Use with Lesson 7-3
GI
discover multiplication properties
of exponents.
<D
@<I
Activity 1
Online Edition
1 Copy and complete the table below.
3 2 3 3 = (3 · 3)(3 · 3 · 3) = 3 5
Countdown Week 15
54 52 = (
Resources
Algebra Lab Activities
5 )( 5
5 )= 56
4 )= 4
5
5
43 43 = ( 4 4
4 )( 4
4
23 22 = ( 2 2
2 )( 2
2 )= 2
63 64 = (
7-3 Lab Recording Sheet
5
)(
6
5
) = 6 · 6 · 6; 6 · 6 · 6 · 6; 6 7
Teach
2 Examine your completed table. Look at the two exponents in each factor and the
exponent in the final answer. What pattern do you notice?
Discuss
3 Use your pattern to make a conjecture: a m · a n = a
The exp. in the final answer is the sum of the exponents in the orig. problem.
. m+n
Remind students that a conjecture is
an educated guess that is based on
evidence but has not been proven
true or false.
Try This
Use your conjecture to write each product below as a single power.
2. 7 2 · 7 2 7 4
3. 10 8 · 10 4 10 12
1. 5 3 · 5 5 5 8
Discuss with students what patterns
they notice in each activity. Help
students represent the patterns in
conjectures with words and with
algebraic statements. For example,
Activity 3 can be thought of as “distributing” the exponent.
4. 8 7 · 8 3 8 10
5. Make a table similar to the one above to explore what happens when you multiply
more than two powers that have the same base. Then write a conjecture in words
to summarize what you find.
Activity 2
1 Copy and complete the table below.
(2 3)2 = 2 3 2 3 = ( 2
2
(2 2)3 = 2 2 2 2 2 2 = ( 2
(4 )
2 4
=
2 )( 2
2 )( 2
42 42 42 42 = ( 4
(3 4)2 = 3 4 3 4 = ( 3
3
3
2
2 )=26
2 )( 2
2 )= 2
6
4 )( 4
4 )( 4
4 )( 4
3 )( 3
3
3 )= 3
3
4 )= 4
8
8
(6 3)4 = 6 3 · 6 3 · 6 3 · 6 3 = (6 · 6 · 6)(6 · 6 · 6)(6 · 6 · 6)(6 · 6 · 6) = 6 12
2 Examine your completed table. Look at the two exponents in the original
expression and the exponent in the final answer. What pattern do you notice?
The exp. in the final answer is the product nof the exponents in the orig. problem.
3 Use your pattern to make a conjecture: (a m) = a . mn
472
Chapter 7 Exponents and Polynomials
3 · 33 · 34 = (3 · 3) · (3 · 3 · 3) · (3 · 3 · 3 · 3) = 39
A1NL11S_c07_0472-0473.indd 2472
22 · 22 · 23 = (2 · 2) · (2 · 2) · (2 · 2 · 2) = 27
42 · 45 · 43 = (4 · 4) · (4 · 4 · 4 · 4 · 4) · (4 · 4 · 4) = 410
62 · 63 · 63 · 62 = (6 · 6) · (6 · 6 · 6) · (6 · 6 · 6) · (6 · 6) = 610
When multiplying powers with the same base, the answer
is the base raised to the sum of all the exponents.
KEYWORD: MA7 Resources
472
Chapter 7
6/25/09 9:08:21 AM
Close
Try This
Key Concepts
Use your conjecture to write each product below as a single power.
4
2
2
6. (5 3) 5 6
7. (7 2) 7 4
8. (3 3) 3 12
The product of two powers with the
same base is the base raised to the
sum of the exponents.
(9 ) 9 21
7 3
9.
10. Make a table similar to the one in Activity 2 to explore what happens when you
raise a power to two powers, for example, ⎡⎣(4 2)
words to summarize what you find.
⎤⎦ . Then write a conjecture in
3 3
The power of a power is the base
raised to the product of the
exponents.
The power of a product is the product of each factor raised to that
exponent.
Activity 3
1 Copy and complete the table below.
Assessment
(ab)3 = (ab)(ab)(ab) = (a a a)(b b b) = a 3 b
(mn)4 = (mn)(mn)(mn)(mn) = ( m m m m )( n
(xy)2 = ( xy )( xy ) = ( x x )( y y ) = x
2
y
Journal Have students use numbers
to show an example of each property explored.
3
n
n
n )= m
4
n
4
2
(cd)5 = (cd )(cd )(cd )(cd )(cd ) = ( c c c c c )( d d d d d ) = c
5
d
5
(pq)6 = pq · pq · pq · pq · pq · pq ; (p · p · p · p · p · p) ·(q · q · q · q · q · q) ; p 6q 6
2 Examine your completed table. Look at the original expression and the final
answer. What pattern do you notice?
To get the final answer, the exp. is “distributed” to each factor in the product.
n
b . n; n
3 Use your pattern to make a conjecture: (ab) = a
Try This
Use your conjecture to write each power below as a product.
7
11. (rs)8 r 8s 8
12. (yz)9 y 9z 9
13. (ab) a 7b 7
14. (xz)12 x 12z 12
15. Look at the first row of your table. What property or properties allow you to write
(ab)(ab)(ab) as (a · a · a)(b · b · b)? Assoc. and Comm. Properties of Mult.
16. Make a table similar to the one above to explore what happens when you raise a
product containing more than two factors to a power, for example, (xyz)7. Then
write a conjecture in words to summarize what you find. Possible answer:
(abc)3 = (abc)(abc)(abc) = (a · a · a)(b · b · b)(c · c · c) = a 3b 3c 3
(xyz)2 = (xyz)(xyz) = (x · x)(y · y)(z · z) = x 2y 2z 2
(mnpq)4 = (mnpq)(mnpq)(mnpq)(mnpq) =
(m · m · m · m)(n · n · n · n)(p · p · p · p)(q · q · q · q) = m 4 n 4 p 4 q 4
When a product is raised to a power, the exp. is “distributed” to each factor in the product.
7- 3 Algebra Lab
473
A1NL11S_c07_0472-0473.indd 473
[(3 ) ]
[(2 ) ]
[(4 ) ]
4
2 3
= (32) · (32) · (32) · (32) = 36 · 36 · 36 · 36 = 324
3
2 2
= (22) · (22) · (22) = 24 · 24 · 24 = 212
3
2 5
= (42) · (42) · (42) = 410 · 410 · 410 = 430
3
2
5
3
2
5
3
3
6/25/09 9:08:28 AM
2
5
When a power is raised to 2 powers, the final answer
is the base raised to the product of all the exponents.
7-3 Algebra Lab
473
7-3
Organizer
Block
Multiplication Properties
of Exponents
7-3
__1 day
2
Objective: Use multiplication
GI
properties of exponents to evaluate
and simplify expressions.
<D
@<I
Online Edition
Who uses this?
Astronomers can multiply expressions with
exponents to find the distance between
objects in space. (See Example 2.)
Objective
Use multiplication
properties of exponents
to evaluate and simplify
expressions.
Tutorial Videos
You have seen that exponential expressions are
useful when writing very small or very large
numbers. To perform operations on these numbers,
you can use properties of exponents. You can also
Countdown Week 15
In this lesson, you will learn some properties that
containing multiplication.
Warm Up
Simplifying Exponential Expressions
Write each expression using an
exponent.
1. 2 · 2 · 2
An exponential expression is completely simplified if…
23
• There are no negative exponents.
2. x · x · x · x
1
1
3. _ 4-2 or _2
4·4
4
Write each expression without
using an exponent.
x4
4·4·4
4. 43
6. m -4
5. y 2
• The same base does not appear more than once in a product or quotient.
• No powers are raised to powers.
• No products are raised to powers.
• No quotients are raised to powers.
• Numerical coefficients in a quotient do not have any common factor other
than 1.
y·y
1
__
m·m·m·m
Examples
b x3
_
a
Also available on transparency
z
12
4
a b
4
Nonexamples
5a 2
s5 _
_
t 5 2b
a
-2
ba x · x 2
(z 3)4 (ab)4
(_st )
5
10a 2
_
4b
Products of powers with the same base can be found by writing each power as
repeated multiplication.
Q: What do you call xsun?
Notice the relationship between the exponents in the factors and the exponent in
the product: 5 + 2 = 7.
A: Solar power.
Product of Powers Property
WORDS
NUMBERS
The product of two powers
with the same base equals
that base raised to the sum
of the exponents.
474
67 · 6 4 = 6 7+4 = 6 11
ALGEBRA
If a is any nonzero real number
and m and n are integers, then
a m · a n = a m+n.
Chapter 7 Exponents and Polynomials
1 Introduce
A11NLS_c07_0474-0480.indd
e x p l o474
r at i o n
7-3
Multiplication Properties
of Exponents
1. The expression x · x · x · x · x can be evaluated two ways.
As a product of two powers, it can be written as x 3 · x 2.
As a single power, it can be written as x 5. Use this
information to complete the table below.
Expression
Product of
Powers
Single
Power
x x x x x x3 x2
x5
y y y y y y a a a a a a a a a m m m m m m 2. Describe any patterns you see in the table above.
3. Use a similar method to complete the table below.
Expression
Product of
Powers
Single
Power
y y y y y y y y3 y2 y2
y7
b b b b b b b b b KEYWORD: MA7 Resources
z z z z z z z z z z x x x x x x x x THINK AND DISCUSS
4. Describe any patterns you see in the second table.
474
Chapter 7
5. Explain how you can use your findings to write x 10 · x 4 as a
single power.
Motivate
Draw a square on the board. Label one side x3.
Ask students for an expression for the area of the
square. x3 · x3
Show students that this can also be written as
(x3)2, and tell them that the properties in this lesson will show them how to simplify expressions
with multiple exponents, such as this one.
Explorations and answers are provided in
Alternate Openers: Explorations Transparencies.
8/18/09 9:54:52 AM
EXAMPLE
1
Finding Products of Powers
Simplify.
A 25 · 26
2 ·2
2 5+ 6
2 11
5
6
2
-2
2
-2
B 4 ·3
C a ·b ·a
5
5
Simplify.
A. 32 · 35
·4 ·3
4 · 3 · 45 · 36
(4 2 · 4 5) · (3 -2 · 3 6)
4 2+5 · 3 -2+ 6
47 · 34
4
Example 1
Since the powers have the same base, keep the
6
Group powers with the same base together.
B. 24 · 34 · 2-2 · 32
Add the exponents of powers with the same base.
C. q3 · r2 · q6
q9r2
D. n3 · n-4 · n
1
Example 2
Group powers with the same base together.
Add the exponents of powers with the same base.
D y 2 · y · y -4
(y 2 · y 1) · y -4
Group the first two powers.
y 3 · y -4
The first two powers have the same base, so add
the exponents.
The two remaining powers have the same base,
y -1
1
_
y
1b. 3 -3 · 5 8 · 3 4 · 5 2 3 × 5 10
1d. x · x -1 · x -3 · x -4 1
_
n4
2
Light from the Sun travels at
about 1.86 × 105 miles per
seconds for the light to reach
Neptune. Find the approximate distance from the Sun to
scientific notation.
2.79 × 109 mi
Write with a positive exponent.
Simplify.
1a. 7 8 · 7 4 7 12
m5
1c. m · n -4 · m 4
EXAMPLE
22 · 36
2
a4 · b5 · a2
(a 4 · a 2)· b 5
a6 · b5
a 6b 5
A number or variable
written without an
exponent actually
has an exponent
of 1.
10 = 10 1
y = y1
37
_
INTERVENTION
x7
Questioning Strategies
EX AM P LE
Astronomy Application
• How do you know when an
expression containing exponents is
completely simplified?
Light from the Sun travels at about 1.86 × 10 5 miles per second. It takes
about 500 seconds for the light to reach Earth. Find the approximate
distance from the Sun to Earth. Write your answer in scientific notation.
• How do you know which powers
to group together?
distance = rate × time
= (1.86 × 10 5) × 500
= (1.86 × 10 5) × (5 × 10 2)
1
Write 500 in scientific notation.
EX AM P LE
2
• What is the formula for distance?
= (1.86 × 5) × (10 × 10
5
)
2
= 9.3 × 10 7
Use the Commutative and
Associative Properties to group.
• Why should you write the time in
scientific notation?
• If you left the time in standard
notation, how would that affect
Multiply within each group.
The Sun is about 9.3 × 10 miles from Earth.
7
2. Light travels at about 1.86 × 10 5 miles per second. Find the
approximate distance that light travels in one hour. Write your
answer in scientific notation. 6.696 × 10 8 mi
7-3 Multiplication Properties of Exponents
475
2 Teach
A1NL11S_c07_0474-0480.indd 475
6/25/09 9:11:42 AM
Guided Instruction
Introduce the Product of Powers Property
by writing each power in the product in
factored form and having students discover
the relationship among the exponents.
Have students discover the other properties
in a similar fashion. In Example 1, encourage students to add the exponents of powers with the same base before rewriting
negative exponents as positive exponents.
Through Graphic Organizers
Have students create a graphic organizer to
show evidence that each property works.
Product of
Powers
Power of a
Power
Power of a
Product
32 · 33
9 · 27
243
(23)4
84
4096
(2 · 4)3
83
512
35
243
212
4096
23 · 43
8 · 64
512
Lesson 7-3
475
To find a power of a power, you can use the meaning of exponents.
Example 3
Notice the relationship between the exponents in the original power and the
exponent in the final power: 3 · 2 = 6.
Simplify.
A. (52)
58
B. ( )
1
C. ( )
· x4
4
0
43
-5
x3
Power of a Power Property
WORDS
1
_
x11
NUMBERS
A power raised to another
power equals that base
raised to the product of the
exponents.
INTERVENTION
EXAMPLE
Questioning Strategies
3
ALGEBRA
If a is any nonzero real
number and m and n are
n
integers, then (a m) = a mn.
(6 7) 4 = 6 7 · 4 = 6 28
Finding Powers of Powers
Simplify.
EX A M P L E
3
A (7 4)
3
74·3
7 12
• What operation is performed on
the exponents when a power is
raised to another power?
Use the Power of a Power Property.
Simplify.
B (3 )
6 0
36·0
30
1
Reading Math Tell students that the exponent is
followed by the word power. For
example, 74 is read “seven to the
fourth power.” Point out that 2 and 3
are exceptions to this rule. For those
exponents, it is customary to say
“squared” or “cubed,” respectively. If
students have trouble with ordinals,
suggest that they make a chart with
the numbers in the first
ENGLISH
column and their ordinals LANGUAGE
LEARNERS
in the second column.
Use the Power of a Power Property.
Zero multiplied by any number is zero.
Any number raised to the zero power is 1.
C (x )
2 -4
·x
5
x 2 · (-4) · x 5
x
-8
x
·x
Use the Power of a Power Property.
5
Simplify the exponent of the first term.
-8 + 5
Since the powers have the same base, add the
exponents.
x -3
1
_
x3
Write with a positive exponent.
Simplify.
3a. (3 4)
5
3 20
3b. (6 0)
3
3c. (a 3) · (a -2)
4
1
-3
a 18
Multiplication Properties of Exponents
Sometimes I can’t remember when to
add exponents and when to multiply
them. When this happens, I write
everything in expanded form.
Briana Tyler
Memorial High School
476
Chapter 7
3
Then (x 2) = x 2 · 3 = x 6.
3
This way I get the right answer even
if I forget the properties.
Chapter 7 Exponents and Polynomials
A11NLS_c07_0474-0480.indd 476
476
For example, I would write
x 2 · x 3 as (x · x)(x · x · x) = x 5.
Then x 2 · x 3 = x 2 + 3 = x 5.
I would write (x 2) as x 2 · x 2 · x 2,
which is (x · x)(x · x)(x · x) = x 6.
12/11/09 9:25:56 PM
Powers of products can be found by using the meaning of an exponent.
""
Ê,,",
,/
when a power is raised to another
power. Encourage students to write
out the expression in expanded form
to help them remember.
Power of a Product Property
WORDS
NUMBERS
(2 · 4) = 2 · 4
3
A product raised to a power
equals the product of each
factor raised to that power.
EXAMPLE
4
3
ALGEBRA
If a and b are any nonzero
real numbers and n is any
n
integer, then (ab) = a nb n.
3
= 8 · 64
= 512
Communicating Math
Have students write the
Product of Powers, Power
of a Power, and Power of a Product
Properties in their own words. Then
aloud.
Finding Powers of Products
Simplify.
A (-3x)2
(-3)2 · x 2
Use the Power of a Product Property.
9x 2
Simplify.
B - (3x)2
In Example 4B, the
negative sign is not
part of the base.
-(3x)2 = -1 · (3x)2
- (3 2 · x 2)
- (9 · x 2)
-9x 2
C
x
·y
x -6 · y 0
x -6 · 1
1
_
x6
Simplify.
Simplify.
(x -2 · y 0)3
(x -2)3 · (y 0)3
-2 · 3
Example 4
Use the Power of a Product Property.
2
B. (-2y)
3
-4y2
-8y3
2
x12
_
C. (x6 · y-3)
y6
Use the Power of a Product Property.
0·3
A. -(2y)
Use the Power of a Power Property.
Simplify.
Write y 0 as 1.
Write with a positive exponent.
INTERVENTION
Questioning Strategies
Simplify.
4a. (4p)3 64p 3
4b. (-5t
) 25t
2 2
4
4c. (x y
) · (x y )
2 3 4
2 4 -4
EX AM P LE
1
_
• How does the Power of a Product
Property compare with the
Distributive Property?
y4
THINK AND DISCUSS
1. Explain why (a
)
2 3
4
• If a negative number is raised to an
even-numbered power, is it positive or negative? Why?
and a 2 · a 3 are not equivalent expressions.
2. GET ORGANIZED Copy and complete the graphic organizer. In each box,
supply the missing exponents. Then give an example for each property.
ÕÌ«V>ÌÊ*À«iÀÌiÃÊvÊÝ«iÌÃ
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*ÜiÀÃÊ*À«iÀÌÞ
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*ÜiÀÊ*À«iÀÌÞ
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Ê ÊÊ®ÊÊÊÊ>Ê Ê
Ê­>L®ÊÊÊÊ>Ê ÊLÊ Ê
7-3 Multiplication Properties of Exponents
3 Close
A11NLS_c07_0474-0480.indd 477
Summarize
Have students identify the property they
can use to simplify each expression. Then
have them simplify the expression.
(32)3
Power of a Power; 36
x2 · x4
Product of Powers; x6
16b2
Power of a Product; _
c4
(4b · c-2)2
477
1. (a2) = a2·3 = a6, while
a2 · AM
a3 = a2+3 = a5.
8/18/09 10:13:30
3
and INTERVENTION
2. See p. A6.
Diagnose Before the Lesson
7-3 Warm Up, TE p. 474
Monitor During the Lesson
Check It Out! Exercises, SE pp. 475–477
Questioning Strategies, TE pp. 475–477
Assess After the Lesson
7-3 Lesson Quiz, TE p. 480
Alternative Assessment, TE p. 480
Lesson 7-3
477
7-3
7-3 Exercises
Exercises
KEYWORD: MA7 7-3
KEYWORD: MA7 Parent
GUIDED PRACTICE
Assignment Guide
SEE EXAMPLE
Assign Guided Practice exercises
as necessary.
2. 5 3 · 5 3 5 6
Simplify.
SEE EXAMPLE
6.
3
p. 476
9.
SEE EXAMPLE 4
p. 477
4. x 2 · x -3 · x 4 x 3
3. n 6 · n 2 n 8
5. Science If you traveled in space at a speed of 1000 miles per hour, how far would
you travel in 7.5 × 10 5 hours? Write your answer in scientific notation. 7.5 × 10 8 mi
2
p. 475
If you finished Examples 1–4
Basic 18—65, 68, 75, 80—83,
98—106
Average 18—43, 44—52 even,
53, 54, 56—64 even,
66—83, 84—96 even,
98—106
56—106
Quickly check key concepts.
Exercises: 18, 22, 28, 32, 40, 42,
46
Simplify.
1. 2 2 · 2 3 2 5
SEE EXAMPLE
If you finished Examples 1–2
Basic 18—22, 53, 68
Average 18—22, 53, 68, 84
Homework Quick Check
1
p. 475
(x 2) 5 x 10
(3 -2) 2 1 , or 1
4
_ _
10.
81
3
12. (2t)5 32t 5
15.
7.
(-2x ) -8x 15
5 3
(y 4) 8
y 32
(a )
-3 4
· (a
(p 3) 3
8.
p9
11. xy · (x 2) · (y 3)
) a2
3
7 2
4
x 7y 13
2
13. (6k) 36k 2
14.
(r 2s) r 14s 7
16. - (2x
17.
(a 2b 2)5 · (a -5)2 b 10
) -8x 15
5 3
7
PRACTICE AND PROBLEM SOLVING
18–21
22
23–28
29–34
x
_
Simplify.
Independent Practice
For
See
Exercises Example
18. 33 · 2 3 · 3 2 3 · 3 4 19. 6 · 6 2 · 6 3 · 6 2 6 8 20. a 5 · a 0 · a -5 1
1
2
3
4
21. x 7 · x -6 · y -3 y 3
22. Geography Rhode Island is the smallest state in the United States. Its land area
is about 2.9 × 10 10 square feet. Alaska, the largest state, is about 5.5 × 10 2 times
as large as Rhode Island. What is the land area of Alaska in square feet? Write your
answer in scientific notation. 1.595 × 10 13 ft 2
Extra Practice
_1
Simplify.
Skills Practice p. S16
(2 3)3 2 9, or 512
6
26. (b 4) · b b 25
23.
Application Practice p. S34
29. (3x)
3
32.
27x
24.
27.
3
30.
(-4x ) 256x 12
3 4
33.
(3 6)0 1
12
4
3 a
b · (a 3) · (b -2)
b5
(5w 8)2 25w 16
4
- (4x 3) -256x 12
(x 2)-1 2
x
(x 4)2 · (x -1)-4 x 12
25.
_
28.
34.
(p 4q 2) 7 p 28q 14
(x 3y 4)3 · (xy 3)-2
(a 2b )4 = a 8b 12 3
31.
x 7y 6
Find the missing exponent in each expression.
35. a a 4 = a 10 6
1
38. (a 3b 6) = _
-3
a 9b 18
36.
(a )4 = a 12 3
37.
39.
1
(b 2)-4 = _
8
40. a · a 6 = a 6 0
b
Geometry Write an expression for the area of each figure.
2x 3 42.
41.
ÓÝ
a 2b 4
43.
2m 10n 6
Ê­®ÊÓ
Î
ÊLÊ Ê
Ê
­ÓÊÊ{
Ê ÊÓ®Ê ÓÊ
Ê>ÊÓÊLÊ Ê Ê
ÊÝÊÓÊ
Simplify, if possible.
44. x 6y 5 x 6y 5
47.
45.
(5x 2)(5x 2) 2 125x 6
48.
7-3
Use a Table
To multiply
powers with the
same base,
keep the base
exponents.
A11NLS_c07_0474-0480.indd 478
Product
of a
Power
To find the
power of a
power, keep the
base and
multiply the
exponents.
Power
of a
Power
To find the
power of a
product, apply
the exponent
to each factor.
Power
of a
Product
3
6
y6
(x 2y 2)2(x 2y)-2y 2
52.
Multiplication Properties of Exponents
3
y7
x−5 • x4 • y7 = x−5 + 4 • y7 = x−1y7 =
x
(4 • 4 • 4) (4 • 4 • 4 • 4 • 4)
48
2 −6
(x ) = x
2 • −6
=x
−12
1
= 12
x
(5 • 3)2 = 52 • 32 = 25 • 9 = 225
(−4b)3 = (−4)3 • b3 = −64b3
(x0 • y−1)5 = (x0)5 • (y−1)5 = x0 • y−5 =
1
y5
In general,
(ab)n = anbn
a4 + (−2) • b5
43 + 5
a2 • b5
(a ) = a
88
(a ≠ 0, m and n are integers.)
Simplify (x5)4 • y.
x5 • 4 • y
3•2
x20y
26
multiply
Simplify.
2. 8−2 • 53 • 86
1. 23 • 24
27
8
84 • 53
4
10
7. (3v5)2 9v
10. −(4y7)2 −16y
14
7
4. m • n • m
m 15 n 4
7. (5−3)3 • 40
6. 84 • 84
mn
Simplify (23)2.
2
9. (2 • 9)3 5832
a2b5
m n
(23)2
Simplify each expression.
c
(43) (45)
You can use the Power of a Power Property to find a power raised to another power.
Power of a Product; with both properties, a number is applied
to all parts.
5
Simplify a4 • b5 • a−2.
Simplify (43) (45).
48
4. 3(x + 4) = 3x + 12 shows how the Distributive Property of Multiplication is used to simplify
an expression. Which property of exponents is similar to the Distributive Property? Why?
d8
12/10/09 8:47:00 PM
Count the number of factors.
The number of factors is the exponent.
am • an = am + n (a ≠ 0, m and n are integers.)
In general,
(am)n = am • n
Power of a Product
24
Expand each factor.
Or you can use the Product of Powers Property:
(43)5 = 43 • 5 = 415
(89)0 = 89 • 0 = 80 = 1
What do you do with the exponents to simplify 64 • 68?
5. (m3)8 m
5
Simplify (4 ) (4 ).
In general,
am • an = am + n
3. What is the name of the property that can be used to simplify (6t)−9?
8. c−2 • d 8 • c−3
7-3 RETEACH
7-3
You can multiply a power by a power by expanding each factor.
(43) (45)
43 • 45 = 43 + 5 = 48
38 • 56 • 3−6 = 38 + (−6) • 56 = 32 • 56
1. What do you do with the exponents to simplify (c4)−2?
2.
Review for Mastery
LESSON
Complete each of the following.
Chapter 7
0
_1
Chapter 7 Exponents and Polynomials
LESSON
478
49. a · a · 3a 3a
3
1000
The table below summarizes the multiplication properties that are needed
to simplify expressions with powers.
KEYWORD: MA7 Resources
46. x 2 · y -3 · x -2 · y -3
3
51. 10 2 · 10 -4 · 10 5 10 , or
3
-2
50. (ab) (ab) ab
478
(2x 2) 2 · (3x 3) 3 108x 13
4
4
- (x 2) (-x 2) -x 16
4 2
5. (6 )
3. 24 • 35 • 28 • 3−2
212 • 33
6. (4−3)2
1
68
46
8. (x2)−4 • y−3
9. (u5)−2 • (v3)4
1
1
v 12
59
x8 y3
u 10
Astronomy The graph shows the
approximate time it takes light from the
Sun, which travels at a speed of 1.86 × 10 5
miles per second, to reach several planets.
Find the approximate distance from the
Sun to each planet in the graph. Write
(Hint: Remember d = rt.)
Sunlight Travel Time
to Planets
Earth
Planet
53.
53. Earth:
9.3 × 10 7 mi; Mars:
1.4136 × 10 8 mi;
Jupiter:
4.836 × 10 8 mi;
Saturn:
8.928 × 10 8 mi
b.
Simplify.
59. 15m 12n 9
59.
62.
2600
Jupiter
4800
Saturn
0
1000 2000 3000 4000 5000
Time (s)
/////ERROR ANALYSIS///// Explain the error in each simplification below. What is the
a. x 2 · x 4 = x 8
56.
760
Mars
54. Geometry The volume of a rectangular
prism can be found by using the formula
V = wh where , w, and h represent the
length, width, and height of the prism.
Find the volume of a rectangular prism
whose dimensions are 3a 2, 4a 5, and 4a 2b 2. 48a 9b 2
55.
500
15
_
(-3x 2)(5x -3) - x
(3m 7)(m 2n)(5m 3n 8)
(2 2)2(x 5y)3 16x 15y 3
(x 4)5 = x 9
c.
a
_
(x 2)3 = x 2
3
= x8
7
57. (a 4b)(a 3b -6) b 5
-2
5
16
60. (b 2) (b 4) b
2
63. (-t)(-t) (-t 4) t 7
58.
(6w 5)(2v 2)(w 6) 12v 2w 11
61. (3st)2t 5 9s 2t 7
64.
(2m 2)(4m 4)(8n)2
Exercise 53, you can
explain to students that a
planet’s distance from the Sun varies
at different points in its orbit. This is
because the planets have elliptical
orbits. At a point called perihelion,
a planet is closest to the Sun. At
aphelion, it is farthest from the
Sun. The root in both words, helion,
comes from the Greek word for Sun.
Exercise 75 involves
using the formula for
the speed of light.
This exercise prepares students for
the Multi-Step Test Prep on page 494.
55a. Exponents are multiplied but
should be multiplied; x20.
512m 6n 2
65. Estimation Estimate the value of each expression. Explain how you estimated.
a. ⎡⎣(-3.031) 2⎤⎦
3
b.
c. Exponent is written as a power
but should be multiplied; x6.
(6.2085 × 10 2) × (3.819 × 10 -5)
66. Physical Science The speed of sound at sea level is about 344 meters per second.
The speed of light is about 8.7 × 10 5 times faster than the speed of sound. What is
the speed of light in meters per second? Write your answer in scientific notation and
in standard form. 2.9928 × 10 8 m/s; 299,280,000 m/s
67. Yes; because of
3
2
67. Write About It Is (x 2) equal to (x 3) ? Explain.
the Comm.
Prop. of Mult.,
68. Biology A newborn baby has about 26,000,000,000 cells. An adult has about 1.9 × 10 3
they are both
times as many cells as a baby. About how many cells does an adult have? Write your
equal to x 6.
answer in scientific notation. 4.94 × 10 13
65a. 93, or 729; possible answer:
Round -3.031 to -3.
6.2085 to 6 and round 3.819
to 4.
75c. Assoc. and Comm. Properties
of Mult.
Simplify.
2
69. (-4k) + k 2 17k 2
70. -3z 3 + (-3z)3 -30z 3
71.
72. (2r)2s 2 + 6(rs)2 + 1
2
73. (3a)2b 3 + 3(ab) (2b)
74.
15a 2b 3
10r s + 1
2 2
(2x 2)2 + 2(x 2)2 6x 4
(x 2)(x 2)(x 2) + 3x 2
x 6 + 3x 2
75. This problem will prepare you for the Multi-Step Test Prep on page 494.
a. The speed of light v is the product of the frequency f and the wavelength
w (v = fw). Wavelengths are often measured in nanometers. Nano means 10 -9,
so 1 nanometer = 10 -9 meters. What is 600 nanometers in meters? Write your
-7
m
answer in scientific notation. 6 × 10
b. Use your answer from part a to find the speed of light in meters per second if
f = 5 × 10 14 Hz. 3 × 10 8 m/s
c. Explain why you can rewrite (6 × 10 -7) (5 × 10 14) as (6 × 5) (10 -7) (10 14).
7-3 PRACTICE A
7-3 PRACTICE C
Practice B
LESSON
7-3 Multiplication Properties of Exponents
479
7-3
7-3 PRACTICE B
Multiplication Properties of Exponents
Simplify.
4
LESSON
7-3
Problem Solving
7-3 PROBLEM SOLVING
Multiplication Properties of Exponents
1. In the mid-nineteenth century, several
landowners in Australia released
domestic rabbits into the wild. Suppose
100 rabbits were released. By 1950, the
times. Determine the wild rabbit
A1NL11S_c07_0474-0480.indd 479
population in 1950.
3. Saturn’s smallest moon, Tethys, has a
diameter of about 6.5 102 miles. The
diameter of Jupiter’s largest moon,
Ganymede, is 5 times that of Tethys.
Determine the diameter of Ganymede.
in scientific notation.
2. Barnard’s star is the fifth closest star to
the Earth, after the Sun and the stars in
the Alpha Centauri system. It takes
1.86 108 seconds for light from
Barnard’s star to reach the Earth. Light
travels at a speed of 1.86 105 miles
per second. Calculate the distance from
Barnard’s star to the Earth.
3.46 1013 miles
4. Delaware and Montana have roughly
the same population. Delaware’s area
is 2.49 103 square miles. Montana is
59 times larger. Determine the area of
form and in scientific notation.
147,000 sq mi or
3.25 103 mi
1.47 105 sq mi
5. The formula for the volume of a cylinder
is V = 2r 2 h where r is the radius and h
is the height. What is the volume of the
cylinder shown below?
6. What is the volume of the cube shown
below?
C 24x2y
B 12xy2
D
36xy2
7. Belize borders Mexico and Guatemala
in Central America. It has an area of
2.30 104 square kilometers. Russia
borders fourteen countries and is
7.43 102 times larger than Belize.
What is the area of Russia?
A 1.71 106 sq km C 1.71 108 sq km
B
1.71 107 sq km D 1.71 109 sq km
7-3
Challenge
1. 3 • 3
36 or 729
7-3 CHALLENGE
Using Exponents to Understand Multiplication of Decimals
−6
4. q • q
When you learned how to multiply one decimal by another, you learned to
count decimal places and move the decimal point that many places to the left.
0.003 × 0.02
Write 6 and move the decimal point 5 places to the left.
0.00006
Using properties of exponents, you can understand why this rule works.
1. 0.06 × 0.002
3. 0.15 × 0.0006
0.00012
0.00009
2. 0.04 × 0.012
4. 0.09 × 0.00012
0.00048
0.0000108
You can also find the product 0.003 × 0.02 by using a property of
exponents that you learned. Notice that the final answer shown below
agrees with the answer obtained by applying the rule for multiplication
shown in the example above.
0.003 × 0.02 =
5. 0.06 × 0.002
0.00012
6. 0.04 × 0.012
0.00048
0.00009
8. 0.09 × 0.00012
0.0000108
G 12n9
H
64n9
J 256n9
6 0
10. (w )
0.000006
Multiply the decimals in the same way as the whole numbers.
Count the number of decimal places in each of the three
numbers. Find the sum of those numbers. Move the decimal
point that many places to the left.
2. 25 • 24
3. 23 • 25 • 21
29 or 512
5. r
−3
4
•r •s
−4
1
j4
9. (g4)−2
8. (h2)5
1
h10
2 5
11. (v ) • v
f 18
16. (−5k)2
25k 2
s20 t 9
29 or 512
6. j −2 • j −4 • j 2
r
s4
g8
12. (w5)−2 • w −3
4
1
v 14
1
19. (s4 t)3 • (s4 t 3) 2
Find each product by using a property of exponents.
10. 0.04 × 0.05 × 0.003
7. c5 • b−2 • c3
1
7. 0.15 × 0.0006
0.00000036
1
q7
13. (f 6 )−4 • (f −2)−3
3
2
3
2
3×2
3×2
6
6
×
=
=
=
=
0.00006
×
=
10,000
1000 100 103 102
103 × 102
103 + 2
105
9. 0.06 × 0.002 × 0.003
−1
c8
2
7/18/09 4:56:00 bPM
Find each product by counting decimal places and moving the
decimal point.
11. Write an extension of the rule for multiplying two decimals between 0 and 1 that applies to
multiplying three such decimals.
F 12n6
A 12xy
LESSON
2
w 13
14. (a−2)−3 • (a5)2
a16
15. (3b)4
81b 4
18. (−3p)−2
17. −(4m)3
1
−64m 3
9p 2
20. (a2 b4)2 • (a−2 b3)−1 • a4
21. (x3 y2)−4 • (x2 y−3)−2
1
a10 b 5
x 16 y 2
22. The pitch of a sound is determined by the number of
vibrations produced per second. The note “middle C”
produces 2.62 × 102 vibrations per second. If a pianist
plays middle C for 5 × 10−1 seconds, how many vibrations
will occur?
1.31 × 102 or 131 vibrations
8. In 1989, Voyager 2 discovered six moons
that orbit Neptune. The smallest of
these is Naiad, which orbits Neptune in
a brief 7.2 hours, or 8.22 104 years.
Neptune’s orbit of the Sun takes 2 105
times longer than Naiad’s. How long
does Neptune’s orbit take?
F 10.2 years
H 102 years
G 16.4 years
J
164 years
Lesson 7-3
479
Students who chose
B for Exercise 80 may
have recalled that any
nonzero number raised to the zero
power is one, but they may have forgotten to multiply by x2.
Critical Thinking Rewrite each expression so that it has only one exponent.
(Hint: You may use parentheses.)
76. c 3d 3
In Exercise 82, 4 must be cubed
because it is a factor of an expression that is being cubed. Students
can eliminate choices A and B
immediately because 43 is 64.
(cd )3
77. 36a 2b 2
(_2ab )
3
8a 3
78. _
b3
(6ab)2
k -2
79. _
4m 2n 2
80. Which of the following is equivalent to x 2 · x 0?
0
1
x2
1
(_
2kmn )
2
x 20
5 2
81. Which of the following is equivalent to (3 × 10 ) ?
7
9 × 10 7
9 × 10 10
6 × 10
6 × 10
10
82. What is the value of n 3 when n = 4 × 10 5?
1.2 × 10 9
1.2 × 10 16
6.4 × 10 9
6.4 × 10 16
83. Which represents the area of the triangle?
6x 2
7x 2
12x 2
ÎÝ
24x 2
{Ý
CHALLENGE AND EXTEND
Simplify.
(3 2) 3 2x
x
84. 3 2 · 3 x 3 2 + x
85.
87. (x + 1)-2(x + 1)3 x + 1
88. (x + 1)2(x + 1)-3
90. (4 )
91. (x )
x x
4x
x x
2
xx
1
_
x+1
2
2
86. (x yz) x 2yz 2
3
89. (x y · x z) x 3y + 3z
92. (3x)2y 9 y x 2y
Find the value of x.
93. 5 x · 5 4 = 5 8 4
94. 7 3 · 7 x = 7 12 9
3
95. (4 x) = 4 12 4
96.
(6 2)
x
= 6 16 8
97. Multi-Step The edge of a cube measures 1.2 × 10 -2 m. What is the volume of the
cube in cubic centimeters? 1.728 cm 3
Journal
Have students explain how parentheses affect the expressions ab3c2,
(ab3c)2 and (ab)3c2.
SPIRAL REVIEW
Find the value of x in each diagram. (Lesson 2-8)
98. □ABCD ∼ □WXYZ 100
99. ABC ∼ RST 15
-
7
Have students work in small groups
to create a presentation that explains
the properties learned in this lesson. The presentation should include
visual aids as needed.
ÝÊV
ÎÊV
Ó{ÊvÌ
nÊvÌ
{ÊV
< ÇxÊV 9
xÊvÌ
,
ÝÊvÌ
/
Determine whether each sequence appears to be an arithmetic sequence. If so, find
the common difference and the next three terms. (Lesson 4-6)
100. 5, 1, -3, -7, …
7-3
8
101. -3, -2, 0, 3, … no
yes; d = -4; -11, -15, -19
103. 7,800,000
Write each number in standard form. (Lesson 7-2)
104. 0.000495
103. 7.8 × 10 6
104. 4.95 × 10 -4
102. 0.4, 1.0, 1.6, 2.2, …
yes; d = 0.6; 2.8, 3.4, 4.0
6,000,000
105. 983 × 10 -1 98.3 106. 0.06 × 10 8
Simplify.
480
Chapter 7 Exponents and Polynomials
36 2. z4 · z -2 · z z3
2
5
1
3. (x3)
x6 4. -(t -3)
-_
t15
3
2
9
_
5. (5g) 125g3 6. (-3f -4)
f 8 A1NL11S_c07_0474-0480.indd
3
-2
1
_
7. (x2y) · (x3y2)
y
1. 32 · 34
8. The islands of Samoa have an
approximate area of 2.9 × 103
square kilometers. The area of
Texas is about 2.3 × 102 times
as great as that of the islands.
What is the approximate area
scientific notation.
6.67 × 105 km2
Also available on transparency
480
Chapter 7
480
6/25/09 9:12:25 AM
7-4
7-4
Division Properties
of Exponents
Organizer
Block
__1 day
2
Objective: Use division
properties of exponents to evaluate
and simplify expressions.
Who uses this?
Economists can use expressions with exponents to
calculate national debt statistics. (See Example 3.)
Objective
Use division properties
of exponents to evaluate
and simplify expressions.
Technology Lab
A quotient of powers with the same base can be found
by writing the powers in factored form and dividing
out common factors.
GI
In Technology Lab Activities
<D
@<I
Online Edition
Tutorial Videos
Countdown Week 15
Notice the relationship between the exponents in the original quotient and the
exponent in the final answer: 5 - 3 = 2.
Quotient of Powers Property
WORDS
The quotient of two nonzero
powers with the same base
equals the base raised to the
difference of the exponents.
EXAMPLE
1
NUMBERS
ALGEBRA
67 = 67-4 = 63
_
64
If a is a nonzero real
number and m and n are
a m = a m - n.
integers, then _
an
Warm Up
Simplify.
1
1
2. 2-3 _
or _
8
23
-3 _
9
v6
_
2
-1
-2
3
)
(
3. 3 · x
4. v w
x
w9
3
3
y
y
_
5. 38 · 3-2 36 6. _
z
z3
Write in scientific notation.
1. (x2)
3
()
Finding Quotients of Powers
Simplify.
A
3 6 = 729
Both 3 6 and 729 are
considered to be
simplified.
C
3
_
8
B
32
38 = 38-2
_
32
= 3 6 = 729
x
_
5
x5
x5 = x5-5
_
x5
= x0 = 1
D
a 5b 9 = _
a 5b 9
_
4
a 4b 4
(ab)
= a5-4 · b9-4
2 3 · 3 2 · 5 7 = 2 3 -1 · 3 2 - 4 · 5 7 - 5
__
2 · 34 · 55
= 2 2 · 3 -2 · 5 2
9
(ab)
4
Simplify.
29 4
1a. _
27
3
2
_
y
1b. _4 1
y y3
8. 0.16 × 107
1.6 × 106
7
2·3 ·5
4
5
2
= ab 5
3 × 10-2
2 ·3 ·5
__
2 ·5
=_
32
4 · 25 = _
100
=_
9
9
= a1 · b5
7. 30 × 10-3
Also available on transparency
a b
_
5
x6
(xy)2
Q: Why does _
refuse to be
x2y2
simplified?
2
n
_
m n
_
3
A: One is the loneliest number.
3
_
16
35 · 24 · 43
1d. _
1c.
2 m
5
34 · 22 · 46
(m ) n
5
4
5
7- 4 Division Properties of Exponents
481
1 Introduce
A11NLS_c07_0481-0487.indd 481
e x p l o r at i o n
7-4
Motivate
Division Properties
of Exponents
x can be
1. The expression __
2
5
Simplified
x
simplified by expanding the powers Expression
Form
5
and dividing out common factors:
3
x
__
x
x 5 _________
x x x x x x x x x 3.
__
x2
2
xx
x
a4
___
Use this information to complete
a2
the table.
y7
__
2. Describe any patterns you see in
y5
the table.
3. You can use a similar method when
the exponent in the denominator is
greater than the exponent in the
numerator. For example,
2
xx
x _________
1 __
1
__
_____
x5 xxxxx xxx x3
x 3. Use this information to
complete this table.
THINK AND DISCUSS
x
__
6
8/18/09 10:15:09 AM
49
Have students write out the factors in _6 .
4
4___
•4•4•4•4•4•4•4•4
4•4•4•4•4•4
simplify this quotient of powers. 43 = 64
Explain that in this lesson they will learn
properties for division of powers.
x5
Expression
2
x
__
x5
m3
___
m7
a2
___
a8
6
x
__
x7
Simplified
Form
x 3
4. Explain how the results in the second table compare to
those in the first table.
10
z
5. Describe how you can use your findings to simplify ___
15
z
without first expanding the powers
Explorations and answers are provided in
Alternate Openers: Explorations Transparencies.
KEYWORD: MA7 Resources
Lesson 7-4
481
EXAMPLE
2
Dividing Numbers in Scientific Notation
Simplify (2 × 10 8) ÷ (8 × 10 5) and write the answer in scientific notation.
Example 1
Simplify.
27
x4
A. _
25, or 32 B. _
x
2
x3
2
d4e3
C. _2 d 2e
(de)
3 · 43 · 55 _
25
D. _
32 · 44 · 53 12
2 × 10
(2 × 10 8) ÷ (8 × 10 5) = _
5
8
8 × 10
You can “split up” a
quotient of products
into a product of
quotients:
a ×_
c
a×c =_
_
b×d
b
d
Example:
3 ×_
3×4 =_
4 =_
12
_
5×7
5
7
35
10 8
2 ×_
=_
8 10 5
Write as a product of quotients.
= 0.25 × 10 8 - 5
Simplify each quotient.
= 0.25 × 10 3
Simplify the exponent.
= 2.5 × 10 -1 × 10 3
Write 0.25 in scientific notation as
2.5 × 10 -1.
The second two terms have the
same base, so add the exponents.
Simplify the exponent.
= 2.5 × 10 -1 + 3
= 2.5 × 10 2
Example 2
Simplify (3 × 1010) ÷ (6 × 106)
and write the answer in scientific
notation. 5 × 103
2. Simplify (3.3 × 10 6) ÷ (3 × 10 8) and write the answer in
scientific notation. 1.1 × 10 -2
Example 3
EXAMPLE
4.408 × 109 dollars in fiscal year
2004—05 on public schools.
There were about 7.6 × 105 students enrolled in public school.
What was the average spending
standard form. \$5800
3
In the year 2000, the United States public debt was about 5.6 × 10 12 dollars.
The population of the United States in that year was about 2.8 × 10 8 people.
What was the average debt per person? Give your answer in standard form.
To find the average debt per person, divide the total debt by the number
of people.
5.6 × 10 12
total debt
__
=_
number of people
2.8 × 10 8
12
= 2 × 10 12 - 8
Simplify each quotient.
= 2 × 10
Simplify the exponent.
4
Write in standard form.
3. In 1990, the United States public debt was about 3.2 × 10 12
dollars. The population of the United States in 1990 was about
2.5 × 10 8 people. What was the average debt per person? Write
1
• Why can the Quotient of Powers
Property not be used to simplify
x4
_
?
y3
• Is the smaller exponent always
subtracted from the larger
exponent? Explain.
A power of a quotient can be found by first writing factors and then writing the
numerator and denominator as powers.
2
in scientific notation?
EX A M P L E
Write as a product of quotients.
The average debt per person was about \$20,000.
Questioning Strategies
EX A M P L E
10
5.6 × _
=_
2.8
10 8
= 20,000
INTERVENTION
EX A M P L E
Economics Application
3
• How do you know which number
is the divisor?
Notice that the exponents in the final answer are the same as the exponent in the
original expression.
482
Chapter 7 Exponents and Polynomials
2 Teach
• What mathematical operations are
A11NLS_c07_0481-0487.indd 482
How do you know?
Guided
8/18/09 10:15:51 AM
Instruction
Review the multiplication properties of
exponents because some problems in this
lesson use both multiplication and division
properties. Introduce the division properties in this lesson by examining the powers
written in factored form first.
Through Graphic Organizers
Have students create a graphic organizer to
show evidence that each property works.
279,936
67
Quotient _
=_
1296
of Powers 64
216
Property
Power of
a Quotient
Property
482
Chapter 7
(_35 )
4
= (0.6)
0.1296
4
67-4 = 63
216
34 _
81
_
=
54
625
0.1296
""
Positive Power of a Quotient Property
WORDS
NUMBERS
A quotient raised
to a positive power
equals the quotient
of each base raised
to that power.
EXAMPLE
4
()
3
_
5
4
ALGEBRA
If a and b are
4 nonzero real
3
3
3
3
3
3
·
3
·
3
·
3
_
_
_
_
_
__
= · · · =
= 4 numbers and n is
5 5 5 5
5·5·5·5
5 a positive integer,
()
a
then __
b
n
Ê,,",
,/
an
= ___
.
bn
When using the Power of a Product
Property in Example 4B, students
might forget to apply the power to
the coefficients of terms. Remind
students to apply the power to each
factor in the product, including variables and coefficients.
Finding Positive Powers of Quotients
Simplify.
A
(_34 )
3
(_34 ) = _
4
3
3
Example 4
3
Use the Power of a Quotient Property.
3
27
=_
64
B
(_)
( )
2x 3
yz
2x 3
_
yz
Simplify.
()
Simplify.
4
A. _
7
( )
3
3
(2x 3)
=_
(yz)3
3
3
Simplify.
6
4
( )
3d2
B. _
ef
16
_
49
2
4
81d 8
_
e4f 4
4
_
y6
Use the Power of a Product Property:
(2x 3) 3 = 2 3(x 3)3 and (yz) 3 = y 3z 3.
Simplify 2 3 and use the Power of a Power
3
Property: (x 3) = x 3 · 3 = x 9.
64
2 , or _
_
( )3
2
23
4a. _
32
2x3
C. _3
(xy)
Use the Power of a Quotient Property.
2 3(x 3)
=_
y 3z 3
8x 9
=_
y 3z 3
2
( )
ab 4
4b. _
81
c 2d 3
5
a b
_
5 20
c 10d 15
( )
a 3b
4c. _
a 2b 2
3
INTERVENTION
Questioning Strategies
EX AM P LE
a
_
3
4
• What happened to a variable if it
was in the original expression, but
not in the simplified expression?
b3
1 . What if x is a fraction?
Remember that x -n = _
xn
(_ab )
-n
()
a
1
_
=_
__a n = 1 ÷ b
(b)
n
an
=1÷_
bn
Write the fraction as division.
Use the Power of a Quotient Property.
n
b
=1·_
an
Multiply by the reciprocal.
n
b
=_
an
Simplify.
()
a
b
Therefore, (_
= (_
a) .
b)
b
= _
a
-n
n
Use the Power of a Quotient Property.
n
7- 4 Division Properties of Exponents
A1NL11S_c07_0481-0487.indd 483
Auditory The names of the properties in this lesson sound a lot
alike. Help students differentiate
between the two:
□
Say “Quotient,” and then write _ .
□
x3
Say “of Powers,” and then write _2 .
x
3
Say “Power,” and then write ( ) .
483
6/25/09 9:10:42 AM
Say “of a Quotient,” and then
x 3
write _
y .
()
Lesson 7-4
483
Negative Power of a Quotient Property
WORDS
A quotient raised to a
negative power equals
the reciprocal of the
quotient raised to the
opposite (positive) power.
Example 5
Simplify.
()
3
A. _
4
-3
( )
2x2
B. _
y3
-2
64
_
27
y6
_
4x4
() ( )
2
C. _
3
-2
6m
_
2n
-3
NUMBERS
EXAMPLE
5
()
2
_
3
A
(_25 )
(_25 ) = (_52 )
Questioning Strategies
5
B
• How can you make a negative
exponent positive?
(__ba )
-n
()
= _ba_
n
bn
= ___
an .
Rewrite with a positive exponent.
-3
?
Use the Power of a Quotient Property.
5 3 = 125 and 2 3 = 8.
( )
( ) ( )
3x
_
y2
3x
_
y2
-3
-3
3
y2
= _
3x
Rewrite with a positive exponent.
(y 2)
= _3
(3x)
y6
=_
3 3x 3
y6
=_
27x 3
3
Multiple Representations
Discuss with students the
advantages or disadvantages of different methods of simplifying. For example, in Example 5A,
students could use the Power of a
Quotient Property first:
-3
If a and b are nonzero real
numbers and n is a positive
integer, then
3
5
=_
23
125
=_
8
INTERVENTION
(__25 )
34
=_
24
-3
3
()
4
Simplify.
n3
_
12m3
1
• What is the value of __
5
()
3
= _
2
Finding Negative Powers of Quotients
-3
EX A M P L E
-4
ALGEBRA
-3
53
125
2
=____
= __
= ___
.
-3
8
23
C
5
Use the Power of a Quotient Property.
Use the Power of a Power Property:
(y 2)3 = y 2·3 = y 6.
Use the Power of a Product Property:
(3x) 3 = 3 3x 3.
Simplify the denominator.
2x
(_34 ) (_
3y )
3y
2x
4 _
= (_
(_34 ) (_
3y )
3 ) ( 2x )
-1
-2
-1
-2
1
2
Rewrite each fraction with a positive
exponent.
(3y)2
4 ·_
=_
3 (2x) 2
Whenever all of
the factors in the
numerator or the
denominator divide
out, replace them
with 1.
Use the Power of a Quotient Property.
3 2y 2
4 ·_
=_
3 2 2x 2
1
9 3y 2
4 ·_
=_
2
13
14x
Use the Power of a Product Property:
(3y)2 = 3 2y 2 and (2x) 2 = 2 2x 2.
Divide out common factors.
3y 2
=_
x2
Simplify.
( )
4
5a. _
32
484
-3
9
729
_
, or _
3
43
b c
(b c ) _
16a
2a
64 5b. _
2 3
-4
8 12
4
9s
_t
( ) (_
t ) s
s
5c. _
3
-2
2 -1
4
Chapter 7 Exponents and Polynomials
3 Close
A1NL11S_c07_0481-0487.indd 484
Summarize
and INTERVENTION
Have students give an example of the
Quotient of Powers Property and the
Power of a Quotient Property.
x5
Quotient of Powers: _3 = x5-3 = x2
x
x 3 _
x3
Power of a Quotient : _
y = y3
()
484
Chapter 7
Diagnose Before the Lesson
7-4 Warm Up, TE p. 481
Monitor During the Lesson
Check It Out! Exercises, SE pp. 481–484
Questioning Strategies, TE pp. 482–484
Assess After the Lesson
7-4 Lesson Quiz, TE p. 487
Alternative Assessment, TE p. 487
7/18/09 4:54:37 PM
THINK AND DISCUSS
Quotient of Powers Property and
the Product of Powers Property
require that the bases be the
same. For quotients, you subtract
the exponents. For products,
you add the exponents. In the
Power of a Quotient Property,
each term is raised to the same
power. In the Power of a Product
Property, each factor is raised to
the same power.
1. Compare the Quotient of Powers Property and the Product of Powers
Property. Then compare the Power of a Quotient Property and the Power
of a Product Property.
2. GET ORGANIZED Copy
and complete the graphic
organizer. In each cell,
supply the missing
information. Then
give an example for
each property.
vÊ>Ê>`ÊLÊ>ÀiÊâiÀÊÀi>ÊÕLiÀÃÊ
>`ÊÊ>`ÊÊ>ÀiÊÌi}iÀÃ]ÊÌi°°°


>Ê ÚÚ
ÚÊÊ
L > ÚÚ
>
>Ê
ÚÊÊ
L ÚÚ
2. See p. A7.
7-4
Exercises
7-4 Exercises
KEYWORD: MA7 7-4
KEYWORD: MA7 Parent
GUIDED PRACTICE
SEE EXAMPLE
1
p. 481
SEE EXAMPLE
p. 482
SEE EXAMPLE
_
22 · 34 · 44 2
2. _
29 · 35
3
5 8 25
1. _
56
2
p. 482
(2.8 × 10 11) ÷ (4 × 10 8)
7 × 10 2
6.
p. 483
SEE EXAMPLE
p. 484
4
() _
25
3
16
_
13. (_
4)
9.
2
_
5
2
-2
5
Assign Guided Practice exercises
as necessary.
(5.5 × 10 3) ÷ (5 × 10 8)
7.
1.1 × 10 -5
(1.9 × 10 4) ÷ (1.9 × 10 4)
1
8. Sports A star baseball player earns an annual salary of \$8.1 × 10 6. There are 162
games in a baseball season. How much does this player earn per game? Write your
Simplify.
SEE EXAMPLE 4
_
2
a 5b 6 a
4. _
3 7
a b
b
15x 6 3
3. _
5x 6
Simplify each quotient and write the answer in scientific notation.
5.
3
Assignment Guide
Simplify.
9
10.
14.
( )
x2
_
xy 3
( )
2x
_
y3
3
-4
x
_
3
y
11.
9
_
y 12
16x
15.
4
(( ) )
a3
_
2
a 3b
2
1
_
6 4
a b
If you finished Examples 1–5
Basic 17–45, 47–57, 62–71
Average 17–49, 54–60, 62–71
y 10
y9
12. _
y
y
x
2b 16. _
_
()( ) _
)
(
y
3a
x
2
_
3
-1
3a
_
2b
-2
2
3 -4
2
2
If you finished Examples 1–3
Basic 17–25, 34–37, 47
Average 17–25, 34–37, 47, 59
8
Homework Quick Check
12
Quickly check key concepts.
Exercises: 18, 22, 25, 28, 30, 48
PRACTICE AND PROBLEM SOLVING
Simplify.
3 9 27
17. _
36
5 4 · 3 3 75
18. _
52 · 32
x 8y 3
x5
19. _
x 3y 3
_
3
x 8y 4 y
20. _
9
xz
x yz
Simplify each quotient and write the answer in scientific notation.
21.
23.
Independent Practice
(4.7 × 10 -3) ÷ (9.4 × 10 3) 5 × 10 -7
(4.2 × 10 -5) ÷ (6 × 10 -3) 7 × 10 -3
22.
24.
(8.4 × 10 9) ÷ (4 × 10 -5) 2.1 × 10 14
(2.1 × 10 2) ÷ (8.4 × 10 5) 2.5 × 10 -4
7- 4 Division Properties of Exponents
A1NL11S_c07_0481-0487.indd 485
485
7/18/09 4:54:43 PM
KEYWORD: MA7 Resources
Lesson 7-4
485
For Exercise 47, students
might be interested to
know that the population density of
the United States is actually rather
low. Some cities are dense, but
there is a lot of open land as well. In
comparison, Puerto Rico has a population density of 435 people/km2.
Bermuda’s population density is
even higher, at 1211 people/km2.
Independent Practice
For
See
Exercises Example
17–20
21–24
25
26–29
30–33
1
2
3
4
5
25. Astronomy The mass of Earth is about 3 × 10 -3 times the mass of Jupiter. The
mass of Earth is about 6 × 10 24 kg. What is the mass of Jupiter? Give your answer in
scientific notation. 2 × 10 27 kg
Simplify.
26.
16
(_23 ) _
81
27.
( )
30.
(_17 )
31.
( )
4
Extra Practice
Skills Practice p. S16
Application Practice p. S34
-3
343
3
a4
_
b2
x2
_
y5
a
_
28.
( )
32.
2
_
8w
(_
16 )
w
12
b6
y
_
25
-5
x
10
a
_
29.
( )
33.
196
_
6x
(_14 ) (_
7)
9x
12
6
a 3b 2
_
ab 3
b6
7 -1
7
y
_
3
xy 2
_
x 3y
3
x6
-2
-2
2
Simplify, if possible.
47. 2000: 3 ×
101
1990: 2.65 ×
101
49. Possible answer: The bases are
the same, so subtract the exponent of the denominator from
the exponent of the numerator:
3
(5x 2)
25x 4
38. _
5x 2
27
( ) a
( ) _
a
10
1 44. _
43. (_
)_
( x y ) _1
42.
( )
b -2
_
b3
2
c 2a 3
_
a5
39.
1
_
2
c
_
4
4
2
10 -5 · 10 5
b 10
2 2 -3
-1
x 2y
100
( )
41.
6
-2
-p 4
_
-5p 3
3
3a
_
a3 · a0
40.
_
(3x 3) 3x 5
37. _2
(6x 2) 4
8d 5 2d 2
35. _
4d 3
3
1995: 2.84 × 101
_
2 3
x 2y 3 x y
36. _
a 2b 3 a 2b 3
x6 x
34. _
x5
25
_
p2
(-x 2)
45. _4 -1
- (x 2)
4
y3
46. Critical Thinking How can you use the Quotient of a Power Property to explain
1 = x 0 = x 0 - n = x -n
x0
1
the definition of x -n ? Hint: Think of __
as __
.
xn
xn
(
45
__ = 43 = 64.
2
4
_ _
)
xn
47. Geography Population density is the
number of people per unit of area. The area of
the United States is approximately 9.37 × 10 6
square kilometers. The table shows population
data from the U. S. Census Bureau.
42
When simplifying __
, subtracting
45
the exponents gives a negative
42
1
1
exponent: __
= 4-3 = __
= ___
.
64
45
43
xn
United States Population
Year
Population (to nearest million)
2000
2.81 × 10 8
1995
2.66 × 10 8
Write the approximate population density
1990
2.48 × 10 8
(people per square kilometer) for each of the
given years in scientific notation. Round decimals to the nearest hundredth.
48. Chemistry The pH of a solution is a number that describes the concentration of
hydrogen ions in that solution. For example, if the concentration of hydrogen ions in
a solution is 10 -4, that solution has a pH of 4.
Lemon juice
pH 2
Apples
pH 3
Water
pH 7
Ammonia
pH 11
a. What is the concentration of hydrogen ions in lemon juice? 10 -2
b. What is the concentration of hydrogen ions in water? 10 -7
c. How many times more concentrated are the hydrogen ions in lemon juice than
in water? 10 5, or 100,000, times more concentrated
5
2
4
4
49. Write About It Explain how to simplify __
. How is it different from simplifying __
?
5
2
4
4
Find the missing exponent(s).
7-4 PRACTICE A
7-4 PRACTICE C
________________________________________
___________________
__________________
7-4 PRACTICE B
7-4
486
Division Properties of Exponents
Simplify.
7
6
= 67 – 5 = 6 2 = 36
65
2.
9
3.
4.
7
( )
a2
_
b
4
8
a 3
=_
b 12
( )
x4
_
y
53.
j
j8
5.
§ s3 t ·
8. ¨ 4 ¸
s
1
3
x
–3
7
t
§ 3a ·
10. ¨
¸
–4
27
8
−
81a 4
–2
49t
§ 4s ·
< ¨ ¸
–2
§ 3c ·
13. ¨
¸
2
−
16s 2
–1
§d ·
¨4¸
15. (3.8 × 10 ) ÷ (1.9 × 10 )
2 × 1011
3
81v 4
t4
300,000 yards
18. It takes 5 yards of fabric to manufacture a dress. If the textile
factory turned their entire yearly production of 1.08 × 108 yards
of fabric into dresses, how many could they make? Give your
Chapter 7
–4
16. (2.5 × 10 ) ÷ (5 × 10 )
17. A textile factory produces 1.08 × 108 yards of fabric every
year. If the factory is in operation 360 days a year, what is
the average number of yards of fabric produced each day?
486
45
= 45 – 3 = 42 = 16
43
7-4 RETEACH
7-4
In general,
am
= am –
an
x12 y 4
x11
= x12 – 1 . y 4 – 6 = x 11 y – 2 = 2
xy 6
y
Review for Mastery
LESSON
n
Division Properties of Exponents
The Quotient of Powers Property can be used to divide terms with exponents.
am
= am – n (a 0, m and n are integers.)
an
75
x 7y
Simplify 2 .
Simplify
.
7
x3
75
x7y
5–2
=7
= x7 – 3 • y
72
x3
= 73
To find the
negative power of
a quotient, apply
the positive
exponent to the
numerator and
denominator of the
reciprocal.
Negative
Power
of a
Quotient
–4
4
In general,
n
an
a
= n
b
b
In general,
2
3
–5
5
35 243
3
= = 5 =
32
2
2
x3 y 5 x –4
4
x5 x 20
x8
= 3 = 12 4 = 4
x y
y
x y
a
b
–n
b
a
n
=
bn
= n
a
= x4y
The Positive Power of a Quotient Property can be used to raise quotients to positive
powers.
n
an
a
= n (a 0, b 0, n is a positive integer.)
b
b
3
4
2x 5 Simplify 4 .
y 2
Simplify .
5
3
4
24
2
= 4
5
5
16
=
625
Use the Positive Power of a
Quotient Property.
b13
1. What do you do with the exponents to simplify 8 ? subtract
b
2.
5
Rewrite the expression 8
–4
using a positive exponent.
8
5 1.
56
54
2.
3
Positive Power of a Quotient
4.
12
128
3
10. 2
2
5. 5
144
f 3g 4
7. 7 5
f gh
–3
g3
4
f h
8
27
5
st 8. 4 t 16
625
x y
11. 3 xy 3
t 18
6
s
2
6
x4
y4
9
6. 8
64
81
–2
2c 2 9. 4 c d 5f 12. 7 g 3
5
–2
23
53
32
10
c d
5
g 14
25f 6
x6y 5
y3
23 ( x 5 )3
( y 4 )3
=
8x15
y 12
Use the Power of a
Product Property.
Simplify.
3.
x6 y2
25
2
4. 5
Simplify each expression.
4
=
Use the Positive
Power of a Quotient
Property.
Simplify.
4
4t 3. What is the name of the property that can be used to simplify the expression ?
5
10
2x 5 (2 x 5 )3
4 =
( y 4 )3
y Simplify.
Complete each of the following.
5 × 106
2.16 × 107 dresses
–4
y3
= _ 3; 4
x
Name _______________________________________ Date___________________ Class __________________
5
81m n
16
32
To divide powers
with the same
base, keep the
base and subtract
the exponents.
To find the
32
25
2
positive power of a = 5 =
Positive
3
243
3
quotient, apply the power
exponent to the x 2 y 5 3
( x 2 y 5 )3
x 6 y 15
of a Quotient
=
= x6y12
y =
numerator
( y )3
y3
and denominator. 6
4
3cd 2
–6
A1NL11S_c07_0481-0487.indd 486
4
Simplify. Write the answer in scientific notation.
5
Quotient
of Powers
2
§ § 3mn · –1 ·
14. ¨ ¨
¸
¹
–2
__________________
Use a Table
The table below summarizes the multiplication properties that are needed to
simplify expressions with powers.
§ –t ·
11. − ¨ ¸
16b 4
___________________
7-4
5m 3
( x 4 )2
7.
( x 3 )5
c
_______________________________________
LESSON
20m
4m 2
-1
Chapter 7 Exponents and Polynomials
5
=t
5
j6
c3 d 2
6. 2 5
c d
§6·
12. ¨ ¸
Š
1
w7
d
t 12
= t 12
t7
2
w
w2
§2·
9. ¨ ¸
52.
Practice B
LESSON
1.
x7 = x4 3
51. _
x
x = x2 6
50. _
x4
3
x3 5. 2 y or 8
125
a 7. 2 b 3
6
x 18
x 8. xy 3m 3 6. 2 n 9m 6
y 12
3
a2b 4
3
(ab )
b
a
n4
2
30 9. 20 2
2
7/18/09 4:54:57 PM
54. This problem will prepare you for the Multi-Step Test Prep on page 494.
a. Yellow light has a wavelength of 589 nm. A nanometer (nm) is 10 -9 m.
What is 589 nm in meters? Write your answer in scientific notation. 5.89 × 10 -7 m
b. The speed of light in air, v, is 3 × 10 8 m/s, and v = fw, where f represents the
frequency in hertz (Hz) and w represents the wavelength in meters. What is
the frequency of yellow light? about 5.09 × 10 14 Hz
55. Which of the following is equivalent to (8 × 10 6) ÷ (4 × 10 2)?
2 × 10 3
2 × 10 4
4 × 10 3
( )
12
x
56. Which of the following is equivalent to _
3xy 4
9y 8
_
x 22
Students who chose
C or D for Exercise 55
may have subtracted
4 from 8 instead of dividing.
4 × 10 4
For Exercise 56, remind students
that the exponent outside the parentheses applies to every factor inside
them, including 3.
-2
3y 8
_
x 22
?
3y 6
_
x 12
6y 8
_
x 26
In Exercise 57, point out that the
will be unaffected by the exponent
because it is outside the parentheses.
(-3x)
_
?
4
57. Which of the following is equivalent to
-1
- (3x)4
1
_
81x 4
-81x 4
1
Exercise 54 involves
dividing numbers
written in scientific
notation. This exercise prepares students for the Multi-Step Test Prep on
page 494.
CHALLENGE AND EXTEND
58. Geometry The volume of the prism at right is V = 30x 4y 3.
Write and simplify an expression for the prism’s height in
terms of x and y. 2x 2y 2
xÝÞ
ÎÝ
_
(x + 1) 2
1
60. Simplify _3 .
(x + 1) x + 1
3 2x .
59. Simplify _
3 2x -1
3
61. Copy and complete the table below to show how the Quotient of Powers Property
can be found by using the Product of Powers Property.
Statements
= am · a-n
2.
1
=a ·_
an
m
a
= _
m
3.
4.
Have students compare and contrast
multiplying and dividing two numbers that are each written in scientific notation.
Reasons
+
1. a m-n = a
Journal

an
m; -n
1. Subtraction is addition of the opposite.
2. Product of Powers Property
Have students simplify
3
( )
-3
?
Def. of neg.
exp.
3. −−−−−−−−−−−−−−−−−−−−−−−
(5x2y)
4. Multiplication can be written as division.
step which property or definition
they used.
2xy
· ___
z4
and explain at each
SPIRAL REVIEW
Find each square root. (Lesson 1-5)
6
63. √
1 1
62. √36
64. - √
49 -7
Solve each equation. (Lesson 2-4)
_
7-4
65. √
144 12
Simplify.
_
66. -2(x -1) + 4x = 5x + 3 - 1
67. x - 1 - (4x + 3) = 5x - 1
3
2
Simplify. (Lesson 7-3)
68. 3 2 · 3 3 3 5, or 243 69. k 5 · k -2 · k -3 1
70.
(4t 5) 2 16t 10
71. -(5x 4) -125x 12
3
7- 4 Division Properties of Exponents
_______________________________________
LESSON
7-4
___________________
__________________
Problem Solving
7-4 PROBLEM SOLVING
LESSON
Division Properties of Exponents
7-4
1. Kudzu is a fast-growing vine that has
become a nuisance in the southeastern
United States. It covers 2.5 105 acres
in Alabama. In 2004 the population of
Alabama was estimated to be
6
A1NL11S_c07_0481-0487.indd 4.45
487 10 people. How many acres of
kudzu are there for each person in
Alabama?
0.056 acres
3. Voyager 2 was launched in 1979 to
explore the planets of the outer solar
system. The spacecraft travels an
average of 4.68 106 kilometers
in one year. Determine the speed of
Voyager 2 in kilometers per hour.
(Hint: 1 year = 8760 hours)
5.34 102 km/h
Name _______________________________________ Date___________________ Class __________________
2. A cylindrical water tank has a volume of
6x2y4 cubic meters. The formula for the
volume of a cylinder is r 2h. The water
tank has a radius of xy meters. What is
its height?
6y2 meters
4. The population of Laos is 6.22 106. In
2004 its gross domestic product (GDP)
was \$1.13 1010. The population of
Norway is 4.59 106. In 2004 its GDP
was \$1.83 1011. What is the GDP per
capita, or per person, of Laos and
Norway?
Laos: \$1817;
Norway: \$39,869
6. A storage chest is shaped like a cube.
What is the volume of the storage
chest?
7-4 CHALLENGE
Applying Properties of Exponents to Rational Numbers
You can use the following three facts to discover a new and interesting
• A rational number is the quotient of two integers with a nonzero denominator.
• Every integer can be written as a product of powers of prime numbers, called
the prime factorization of the given number. For example, 120 = 233151.
• When dividing two powers with the same base, subtract the exponents.
1
105
103
= 102 and
=
103
105 102
Write the prime factorization of each integer.
23 31
1. 24
3. 452
22 1131
5.
18
24
6.
48
180
2 • 32 = 3
22 • 3 22
24 • 3 = 22
22 • 32 • 5 3 • 5
7.
250
288
2 • 53 = 53
25 • 32 24 • 32
8.
540
1800
22 • 33 • 5 = 3
23 • 32 • 52 2 • 5
2. 108
4. 1800
22 33
23 32 52
9. Examine the final quotients that you wrote in Exercises 5–8. Explain why a prime-number
base that appears in a numerator does not appear in the denominator and why a primenumber base that appears in a denominator does not appear in the numerator.
A 5b2 yards
C
B 5b3 yards
D 25b6 yards
5b6 yards
7. The wavelengths of electromagnetic
radiation vary greatly. Green light has a
wavelength of about 5.1 107 meters.
The wavelength of a U-band radio wave
is 2.0 102 meters. About how much
greater is the wavelength of a U-band
radio wave than that of green light?
A 2.55 109
C
B 2.55 105
D 3.92 105
3.92 104
F
x3
cubic units
64
H
32
cubic units
x3
x3
cubic units
J 64x 3 cubic units
G
32
8. Puerto Rico has an area of
5.32 103 square miles and a population
of 3.89 106. What is the population
density of Puerto Rico in persons per
square mile?
3
F 1.37 10
G 1.37 102
H
2
7.31 10
J 7.31 103
( )
If a prime number base b appear in the numerator (or
denominator), it cannot occur in the denominator (or
numerator) as well because then the rational number
bna
bn m a
is not fully simplified. ex: m =
c
b c
a
a
10. Let b be a rational number. Write a generalization about the representation of
b
as the quotient of prime numbers raised to powers. Illustrate your generalization by using
21 33
54
120 = 23 31 51 .
Every rational number can be written as a quotient whose numerator
is 1 or the product of prime numbers raised to positive integer
exponents and whose denominator can be written as 1 or the product
of prime numbers raised to positive integer exponents, and there
are no prime bases common to the numerator and the denominator.
8y3
_
x6
x5y2
2. _3
(xy)
x2
_
y
n
( ) _
16m
8c
3 _
9d
_
5. (_
c ) ( 2c )
81d
4m
4. _
n3
2
-2
6
2
2
-3
6
) ÷ (5 ×
6. Simplify (3 ×
105) and write the answer in
scientific notation. 6 × 106
6/25/09 9:10:58 AM
For each rational number, write the numerator and
denominator by using the prime factorization of each. Then use
the Quotient-of-Powers Property to simplify the result. Do not
multiply out the powers of prime numbers that remain.
5. A rectangular parking lot has an area of
10a3b6 square yards. What is the width
of the parking lot?
Challenge
487
48
1. _3 45
4
3
2xy2
3. _
x3y
1012
7. The Republic of Botswana has
an area of 6 × 105 square
kilometers. Its population
is about 1.62 × 106. What
is the population density of
in standard form.
2.7 people/km2
Also available on transparency
Lesson 7-4
487
7-5
Organizer
7-5
Block
Rational Exponents
__1 day
2
Objective: Evaluate and simplify
GI
expressions containing rational
exponents.
<D
@<I
Why learn this?
You can use rational exponents to find the
number of Calories animals need to consume
each day to maintain health. (See Example 3.)
Objective
Evaluate and simplify
expressions containing
rational exponents.
Online Edition
Vocabulary
index
Tutorial Videos, Interactivity
Recall that the radical symbol √ is used to indicate roots. The index is
the small number to the left of the radical symbol that tells which root to take.
3
3
= 2.
For example, √ represents a cube root. Since 2 3 = 2 · 2 · 2 = 8, √8
Countdown Week 16
Another way to write nth roots is by using exponents that are fractions.
= b k.
For example, for b > 1, suppose √b
√
b = bk
( √b )2
Warm Up
5
2. √0
6
3. √
64
4
4. √1
n
When b = 0, √
b = 0.
n
b = 1.
When b = 1, √
0
3
4
So for all b > 1, √
b=
Power of a Power Property
1 = 2k
If b m = b n, then m = n.
1 =k
_
2
Divide both sides by 2.
1
_
b 2.
_1
1
5
5. √100,000
2
b 1 = b 2k
Simplify each expression.
36
1. √
= (b k) Square both sides.
Definition of b n
WORDS
10
NUMBERS
A number raised to the
1
power of __
n is equal to the
nth root of that number.
3
6. - √
27 -3
Also available on transparency
_1
3 2 = √3
_1
4
5 4 = √
5
_1
27
EXAMPLE
and this textbook
Q: What do 3 √7
have in common?
1
= √
2
7
ALGEBRA
If b > 1 and n is an
integer, where n ≥ 2,
_1
n
then b n = √
b.
_1
_1
3
b3 = √
b,
b 2 = √b,
_1
4
and so on.
b 4 = √b,
_1
Simplifying b n
Simplify each expression.
_1
A 125 3
A: Both have an index.
_1
_1
3
3
= √
125 3 = √125
53
=5
√ is equivalent
2
to √. See Lesson
1-5.
_1
_1
_1
_1
Use the definition of b n .
B 64 6 + 25 2
_1
6
64 6 + 25 2 = √
64 + √
25
Use the definition of b n .
6
6
= √2
+ √
52
= 2 + 5 =7
488
Chapter 7 Exponents and Polynomials
1 Introduce
e x p l o r at
A1NL11S_c07_0488_0493.indd
488i o n
Motivate
7-5 Rational Exponents
You will need a graphing calculator for this Exploration.
Recall that whole-number exponents mean repeated
multiplication. For example, 3 4 3 · 3 · 3 · 3 81. You use a radical
4 to show the inverse operation: 兹81 3.
1. You can use your calculator to evalu3 ate radicals. To find 兹4096 , enter 3 and
x then press
. Select 5: 兹 , enter 4096,
and press
.
complete the table.
3 You can also use your calculator to
explore fractional exponents. To evaluate
1
__
4096 3, first enter 4096. Then press
1
3
and press
.
the table.
%.4%2

3 兹4096
4 兹4096
6 兹4096
Power
1
__
1
__
4096 4
1
__
4096 6
THINK AND DISCUSS
Chapter 7
Value
4096 2
4096 3
5. Describe what you notice about the two tables.
1
1
__
Then write 4 2 2 on the board. Ask students to
guess the value of this expression and explain
their thinking. Possible answer: 32; the value
should be between 16 and 64.
Tell students that in this lesson they will learn why
1
__
4 2 2 = 32.
1
__
KEYWORD: MA7 Resources
488
16
Write 42 and 43 on the board and have students
simplify both expressions. 16 and 64
16
Explorations and answers are found in
Alternate Openers: Explorations Transparencies.
7/18/09 4:53:25 PM
""
Simplify each expression.
1
_
1a. 81 4
1
_
1
_
3
1b. 121 2 + 256 4
15
2
_
A fractional exponent can have a numerator other than 1, as in the expression b 3.
You can write the exponent as a product in two different ways.
2
_
1 ·2
_
2
_
b3 = b3
=
b3 = b
( )
1 2
_
b3
= ( √
b)
3
Power of a Power Property
_1
2
1
2·_
3
With fractional exponents with
a numerator other than 1, as in
Example 2, students may confuse
the index with the power. Write
power
_
base index on the board for students
to use as a reference.
1
_
b2 3
=(
Ê,,",
,/
)
2
= √b
3
Definition of b n
m
_
Definition of b n
WORDS
NUMBERS
ALGEBRA
3
) = 2 2 = 4
8 3 = ( √8
If b > 1 and m and n are
integers, where m ≥ 1
and n ≥ 2, then
_2
A number raised to the
m
power of __
n is equal to the
nth root of the number
raised to the mth power.
2
_2
3
3
2 = √
8 3 = √8
64 = 4
b
m
_
n
= ( √
b) =
m
n
Example 1
Simplify each expression.
1
_
m
√
n
A. 343 3
b .
7
1
_
5
1
_
B. 32 + 9 2
2
EXAMPLE
Simplifying Expressions with Fractional Exponents
Simplify each expression.
_2
Example 2
Simplify each expression.
_4
B 32 5
A 216 3
5
_
5
4
A. 81 4
3
3
= √6
= ( √
25)
4
B. 3125
= (6)2 = 36
= (2)4 = 16
_2
m
_
216 3 = ( √
216 )
2
3
(
Definition of b n
)2
_4
32 5 = ( √
32 )
5
2a.
3
EXAMPLE
8
2b.
2
_
15
1
2c.
4
_
27 3
243
2
_
5
25
Example 3
Given a cube with surface area
S, the volume V of the cube can
be found by using the formula
S _3
V = _ 2 . Find the volume of a
6
cube with surface area 54 m2.
27 m3
Simplify each expression.
3
_
16 4
5
81
( )
Biology Application
The approximate number of Calories C that an animal needs each day is
_3
given by C = 72m 4 , where m is the animal’s mass in kilograms. Find the
number of Calories that a 16 kg dog needs each day.
_3
C = 72m 4
_3
= 72(16) 4
Substitute 16 for m.
= 72 · ( √
16 )
4
3
m
_
INTERVENTION
Definition of b n
Questioning Strategies
= 72 · ( √
24)
4
3
= 72 · (2) 3
EX AM P LE
• How do you change a fractional
exponent to an nth root?
= 72 · 8 = 576
The dog needs 576 Calories per day to maintain health.
3. Find the number of Calories that an 81 kg panda needs each day.
1944
7- 5 Rational Exponents
489
EX AM P LE
2
• Will you get the same answer if
you raise the number to the power
before taking the root?
EX AM P LE
2 Teach
3
• How can you use the order of
operations to solve this problem?
A1NL11S_c07_0488_0493.indd 489
7/20/09 4:50:57 PM
Guided Instruction
Review powers and roots by writing
3
53 = 5 • 5 • 5 = 125 and √
125 = 5. Have
students practice writing several of these
examples. Then present the definition of
1
_
bn
1
and discuss Example 1. Show students
1
_
1
_
two special cases: 1 n = 1 and 0 n = 0 for
all natural-number values of n. Continue
m
_
with the definition of b n and the remaining examples. Remind students of the
properties of exponents before presenting
Example 4.
Through Cooperative Learning
Have students work in pairs. Students take
turns rolling both a red (r) and a blue (b)
number cube. After each roll, the student
uses the numbers shown on the cubes to
r
_
complete the expression 64 b . Then the student simplifies the expression or states that
it cannot be simplified. The other student
checks the answer and then rolls the number
cubes to decide the next expression.
Multiple
Representations In
Example 2, the power can
also be placed under the radical
2
_
sign: 216 3 = √
216 2 . However, it
is usually more convenient to
evaluate the root and then
evaluate the power.
3
Lesson 7-5
489
Remember that √ always indicates a nonnegative square root. When you
simplify variable expressions that contain √, such as √
x 2 , the answer cannot
be negative. But x may be negative. Therefore you simplify √
x 2 as ⎪x⎥ to ensure
Example 4
Simplify. All variables represent
nonnegative numbers.
4
4b20
ab5
A. √a
1
_
B. (x6y 4) 2 √
y2
When x is...
and n is...
x n is...
n
n is...
and √x
Positive
Even
Positive
Positive
Negative
Even
Positive
Positive
x3y3
Positive
Odd
Positive
Positive
Negative
Odd
Negative
Negative
n to ⎪x⎥, because you do not know
When n is even, you must simplify √x
n
whether x is positive or negative. When n is odd, simplify √
x n to x.
n
INTERVENTION
Questioning Strategies
EXAMPLE
Using Properties of Exponents to Simplify Expressions
Simplify. All variables represent nonnegative numbers.
4
EXA M PL E
4
A
• How do you use properties of
exponents to simplify expressions?
x 9y 3
√
_1
3
x 9y 3 = (x 9y 3) 3
√
3
_1
Definition of b n
_1
_1
= (x 9) 3 · (y 3) 3
after you have simplified an
expression?
When you are told
that all variables
represent nonnegative numbers,
you do not need to
use absolute values
=
= (x
B
Power of a Product Property
( )·( )
_
_
9· 1
x 3
3· 1
y 3
) · (y ) = x
3
1
Power of a Power Property
3
y
Simplify exponents.
( _) √y
(x y ) √y = (x y_) · y
_
= (x ) · (y ) · y
1 4
3
x 2y 2
1 4
_
3
2 2
3
1 4
2 2
3
Power of a Product Property
) · (y ) · y
8
=x ·y
8
3
1 ·4
2
2·4
= (x
√y3 = y
2
2+1
Simplify exponents.
=x y
8 3
Product of Powers Property
Simplify. All variables represent nonnegative numbers.
4a.
5
_
1 2
_
2
4
x 4y 12 xy 3
√
5
5
√
x
THINK AND DISCUSS
1
_
10
1. ( √
25 ) = 25 10 = 25 2 =
√25
= 5
5
(xy )
4b. _ xy
1. Explain how to find the value of ( √
25 ) .
10
2. GET ORGANIZED Copy and
complete the graphic organizer.
In each cell, provide the definition
and a numerical example of each
type of exponent.
2. See p. A7.
5
Exponent
Definition
Numerical
Example
_1
bn
m
_
bn
490
Chapter 7 Exponents and Polynomials
3 Close
A1NL11S_c07_0488_0493.indd 490
Summarize
Essential Question
Have students give an equivalent expression with fractional exponents for each
expression below. Then have them simplify
the expression.
4
1. √
81
2. ( √
25 )
1
_
81 4 ; 3
3. ( √
256 )
4
490
3
_
3
25 2 ; 125
3
3
_
256 4 ; 64
Chapter 7
Be sure students can answer the lesson’s
essential question:
How do you simplify an expression with a
rational exponent? If the exponent has the
1
form __
n , find the nth root of the base.
m
If the exponent has the form __
n , find the
nth root of the base raised to the mth
power.
and INTERVENTION
Diagnose Before the Lesson
7-5 Warm Up, TE p. 488
Monitor During the Lesson
Check It Out! Exercises, SE pp. 488–490
Questioning Strategies, TE pp. 489–490
Assess After the Lesson
7-5 Lesson Quiz, TE p. 493
Alternative Assessment, TE p. 493
6/25/09 3:18:00 PM
7-5
Exercises
7-5 Exercises
KEYWORD: MA11 7-5
KEYWORD: MA7 Parent
GUIDED PRACTICE
Assignment Guide
5
1. Vocabulary In the expression √
3x, what is the index? 5
Assign Guided Practice exercises
as necessary.
Simplify each expression.
1
_
SEE EXAMPLE
1
3. 16 2 4
6. 81 2 9
1
_
1
_
1
_
2
1
_
1
_
1
_
12. 81 4 + 8 3 5
3
_
17. 25 2 125
2
_
3
_
19. 64 3 256
1
_
13. 25 2 - 1 4 4
16. 125 3 25
4
_
18. 36 2 216
9. 625 4 5
2
_
15. 8 3 32
3
_
p. 489
1
_
5
_
14. 81 4 27
1
_
8. 1 9 1
11. 8 3 + 64 2 10
3
_
SEE EXAMPLE
5. 27 3 3
1
_
7. 216 3 6
10. 36 2 + 1 3 7
1
_
4. 0 6 0
1
_
1
_
p. 488
1
_
1
_
2. 8 3 2
20. 1 4 1
21. 0 3 0
1
_
SEE EXAMPLE
3
p. 489
SEE EXAMPLE 4
p. 490
22. Geometry Given a square with area a, you can use the formula P = 4a 2 to find the
perimeter P of the square. Find the perimeter of a square that has an area of 64 m 2.
32 m
Simplify. All variables represent nonnegative numbers.
23.
27.
x 4y 2 x 2 y
√
(a ) √
a a
1 2
_
2
2
4
24. √z4 z
2
28.
25.
(x ) √y x y
1 6
_
4
3
2
4
x 6y 6 x 3y 3
√
3
12 6
26. √a
b a 4b 2
( )
1 3
_
z3
x 6y 9 3
√
30. _
y
x2
3
29. _ 1
√
z2
Quickly check key concepts.
Exercises: 34, 40, 44, 51, 52, 56
31–42
43–50
51
52–59
1
2
3
4
Simplify each expression.
1
_
32. 1 5 1
1
_
1
_
3
_
Extra Practice
1
_
40. 25 2 - 81 4 2
44. 27 3 9
38. 400 2 20
1
_
1
_
41. 121 2 - 243 5 8
3
_
45. 256 4 64
5
_
3
_
47. 100 2 1000
1
_
37. 256 8 2
2
_
43. 4 2 8
Skills Practice p. S17
1
_
34. 729 2 27
1
_
36. 196 2 14
39. 125 3 + 81 2 14
1
_
33. 512 3 8
1
_
35. 32 5 2
1
_
1
_
1
_
31. 100 2 10
5
_
48. 1 3 1
If you finished Examples 1–4
Basic 32–58 even, 60–80,
85, 87–90, 98–107
Average 32–58 even, 60–90,
92–96 even, 98–107
Homework Quick Check
PRACTICE AND PROBLEM SOLVING
Independent Practice
For
See
Exercises Example
If you finished Examples 1–2
Basic 31–50, 87, 88
Average 32–50 even, 60–78
even, 98–107
even, 86–88, 91–93,
98–107
1
_
1
_
42. 256 4 + 0 3 4
5
_
46. 64 6 32
2
_
49. 9 2 243
50. 243 5 9
Application Practice p. S34
51.. Biology Biologists use a formula to estimate the mass of a mammal’s brain. For
a mammal with a mass of m grams, the approximate mass B of the brain, also in
2
_
grams, is given by B = __18 m 3 . Find the approximate mass of the brain of a mouse that
has a mass of 64 grams. 2g
Simplify. All variables represent nonnegative numbers.
3
6 9
52. √a
c a 2c 3
56.
3
3 2m
53. √8m
( ) √x x y
2
1
_
x 2 y3
2
2 6
57.
1 3
_
a 2b 4 2
(
) √
b 6 ab 4
54.
58.
4
x 16y 4 x 4y
√
3
x 6y 6
√
_
yx 2
3
6 3x 2
55. √27x
(a b )
_
2
y
59.
1 4
_
2
√
b2
a 8b
Fill in the boxes to make each statement true.
_
60. 256 4 = 4 1
64.
_
64 3
= 16 2
61.
1
_
5
65.
3
_
4
=11
= 125 625
1
_
62. 225 = 15 2
4
_
66. 27 = 81 3
63.
1
_
6
67.
_
36 2
=0 0
= 216 3
7- 5 Rational Exponents
A1NL11S_c07_0488_0493.indd 491
491
7/18/09 4:53:46 PM
KEYWORD: MA7 Resources
Lesson 7-5
491
Exercise 85 involves
using rational exponents to model a
real-world situation related to light.
This exercise prepares students for
the Multi-Step Test Prep on
page 494.
Simplify each expression.
81 _12 9
68. _
169
13
( ) _
1
_1
71. (_
16 ) 4
8
16
_
74. (_
81 ) 27
1
1
_
77. (_
81 ) 27
1
_
2
3
_
4
3
_
1
3·_
2
3
_
42
1
_
·3
42
1
_
1
_
= (4 3) 2 = 64 2 = 8;
= (4 )
1
_
2
1
_
3
_2
256
_4
(_
81 ) 3
8
_4
73. (_
27 ) 9
8
_
4
76. (_
25 ) 125
16
8
_
79. (_
125 ) 625
3
27
_
3
_
2
8
_
2
_
3
64
3
_
2
343
9
_
4
_
3
16
80. Multi-Step Scientists have found that the
life span of a mammal living in captivity is
related to the mammal’s mass. The life span
in years L can be approximated by the formula
3
=
= 2 = 8. It
is often easier to take the square
root first so that the remaining
numbers in the calculation are
smaller.
3
Typical Mass of Mammals
Mammal
L=
where m is the mammal’s mass in
kilograms. How much longer is the life span
of a lion compared with that of a wolf?
12 years
Mass (kg)
Koala
8
Wolf
32
1
_
12m 5 ,
101. n < 3
1
_
4
70.
3
_
2
2
_
3
3
_
4
86. 4 2 = 4
8
(_
27 )
9
72. (_
16 )
4
75. (_
49 )
27
78. (_
64 )
69.
Lion
243
Giraffe
1024
1
_
81. Geometry Given a sphere with volume V, the formula r = 0.62V 3 may be used to
approximate the sphere’s radius r. Find the approximate radius of a sphere that has
a volume of 27 in 3. 1.86 in.
102. x ≥ 2
103. y ≤ - 2
104. D: {2}; R: {3, 4, 5, 6}; no; the
domain value 2 is paired with
several different range values.
( _)
1 3
3
82. Critical Thinking Show that a number raised to the power _13_ is the same as the
cube root of that number. (Hint: Use properties of exponents to find the cube
_1 · 3
1
_
82. b
= b3 =
b 1 = b. Also, by
3
3
definition ( √
b) =
3
. Use the fact that if two numbers have
of b 3 . Then compare this with the cube of √b
the same cube, then they are equal.)
2
_
3
_
Thinking Compare n 3 and n 2 for values of n greater than 1. When
_1 383. Critical
simplifying
each of these expressions, will the result be greater than n or less than
b.
b. Therefore, b 3 = √
n?
Explain.
_2
3
_
83. n 3 will be less 84. /////ERROR ANALYSIS///// Two students simplified 64 2 . Which solution is incorrect?
than n because
Explain the error.
__2 < 1. n _2 will be greater
3
than n because __3 > 1.
3
105. D: {-2, -1, -0, 1}; R: {0, 1, 2,
3}; yes; each domain value is
paired with exactly one range
value.
!
?,
2
106. D: {5, 7, 9, 11}; R: {2}; yes; each
domain value is paired with
exactly one range value.
84. Solution A is incorrect.
The first line should be
_3
!
,
"
/-
/-+ Ȗе
!-" +
*/
"
+
?,
"
!
/-+ Ȗе
/-
!1",
.*+
,
64 2 = ( √
64 ) .
3
107. D: 1 ≤ x ≤ 4; R: 2 ≤ y ≤ 4; yes;
each domain value is paired
with exactly one range value.
85. This problem will prepare you for the Multi-Step Test Prep on page 494.
You can estimate an object’s distance in inches from a light source by using the
(
)
_1_
L 2
, where L is the light’s luminosity in lumens and B is the light’s
formula d = 0.8__
B
brightness in lumens per square inch.
a. Find an object’s distance to a light source with a luminosity of 4000 lumens and a
brightness of 32 lumens per square inch. 10 in.
b. Suppose the brightness of this light source decreases to 8 lumens per square inch.
How does the object’s distance from the source change?
7-5 PRACTICE A
The distance doubles (20 in.)
7-5 PRACTICE C
a e ________________________________________
Practice B
LESSON
7-5
a e __________________ C ass__________________
7-5 PRACTICE B
492
Rational Exponents
Chapter 7 Exponents and Polynomials
Simplify each expression. All variables represent nonnegative
numbers.
________________________________________
1
1
1
1. 27 3
2. 1212
3. 0 3
3
4.
11
1
64 2
+
1
27 3
5.
11
1
16 4
1
+ 83
6.
4
1
8. 25
8
8
5
3
11. 16
12. 1212
5
1
y5
14.
1
A number raised
to the power of
1
is equal to the
n
92 = 9 = 3
In general, if b > 1 and n is
an integer where n ≥ 2,
1
32 5
then b n = n b .
1
nth root of that
number.
= 32 = 2
15.
3
Definition
m
of b n
1
1
17. ( x 3 y )3 x 2 y 2
x5
x 2y 4
18.
x3
A number raised to
3
3
m
16 4 = 4 16 = 23 = 8
is
the power of
n
3
3
equal to the nth
100 2 = 100
root of the number
raised to the mth
= 103 = 1000
power.
(
)
In general, if b > 1 and m
and n are integers where
m ≥ 1 and n ≥ 2, then
m
)
bn =
(n b )
m
n
= bm .
1. The expression
1
27 3
is equal to the 3rd
or cube root of 27.
4.
7.
4
32 5
is equal to the 5th root of 32 raised to the 4th power.
5.
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Chapter 7
1
812
1
+ 32 5
1
= 81 + 5 32
=9+2
= 11
1
1
1
1. 64 2
2. 1000 3
3. 15
8
10
6.
9
2
1
400 2
2
32 5
3
814
8.
9.
4
1
1
1
5. 32 5
6. 49 2
4
2
7
1
27
5
3
11. 16 2
12. 256 4
l
1024
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64
t th
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th
ibilit
f th i
t
t
1
1
1
8. 1212 + 27 3
1
3
1
1
11. 144 2 − 125 3
7
1
9. 32 5 + 12
14
1
1
3
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1
10. 814 − 16 4
10. 4 2
t
1
4. 256 4
6
1
16 4
2
8
492
1
1
812
7. 8 3 + 16 2
1
83
20
216 = 6.
Simplify 812 + 32 5 .
5
2. The expression 9 2 is equal to the square root of 9 raised to the 5th power.
i ht © b H lt M D
7/20/09 4:52:27 PM
3
Simplify each expression.
3. The expression
tC
1
63 = 6 × 6 × 6 = 216, so
Simplify each expression.
t
1
, write the nth root of the number.
n
1
216 3 .
Think: What number, when taken as a factor 3 times, is equal to 216?
Complete each of the following.
20 m
l
Simplify
1
x
1
19. Given a cube with volume V, you can use the formula P = 4V 3 to find
the perimeter of one of the cube’s square faces. Find the perimeter of
a face of a cube that has volume 125 m3.
Oi i
Rational Exponents
125 3 = 3 125 = 5
(
(x 4 )8
3
7-5 RETEACH
7-5
When an expression contains two or more expressions with fractional exponents,
evaluate the expressions with the exponents first, then add or subtract.
a 6 b3
a 2b
16. ( x 2 )4 x 6
Review for Mastery
LESSON
216 3 = 3216
=
6
1
5
Name ________________________________________ Date __________________ Class__________________
To simplify a number raised to the power of
1331
x 4 y 12
x 2y 6
y
1
bn
A1NL11S_c07_0488_0493.indd 492
3
8
The table below summarizes the definitions you need when you work with
fractional exponents.
of
9. 32 5
125
Use a Table
Definition
3
3
2
10. 16 4
13.
−
1
64 6
8
1
7. 15 + 49 2
1
100 2
__________________
7-5
0
__________________
LESSON
1
1
12. 625 4 − 0 2
5
3
_
1
3·_
1 ·3
_
86. Write About It You can write 4 2 as 4 2 or as 4 2 . Use the Power of a Power
Property to show that both expressions are equal. Is one method easier than the
other? Explain.
""
In Exercise 80, students might
multiply by 12 before evaluating the
fifth root of m. Remind students of
the order of operations: grouping
symbols, exponents, multiply/divide,
1
_
1
_
87. What is 9 2 + 8 3 ?
4
5
6
10
88. Which expression is equal to 8?
3
_
1
_
4
_
3
_
42
16 2
32 5
64 2
a 3b
a 3b 3
3
9 3
89. Which expression is equivalent to √a
b ?
a 2b
a3
for Exercise 89, they
may think that
3
_
1
_
90. Which of the following is NOT equal to 16 2 ?
)
( √16
3
)
( √16
2
3
43
3
√
b 3 = 1 or (b 3) 3 = 1. Remind
3
√16
them that cubing a number and
taking the cube root are inverse
operations that “undo” each other,
1
_
3
b 3 = (b 3) 3 = b.
so √
CHALLENGE AND EXTEND
Use properties of exponents to simplify each expression.
(a )(a )(a ) a
1
_
3
91.
1
_
3
1
_
3
92.
(x ) (x ) x
1 5
_
2
3
_
2
Ê,,",
,/
4
93.
(x ) (x )
1
_
5 3
1 4
_
3
x3
You can use properties of exponents to help you solve equations. For example, to
_1
_1
1
power to get (x 3) 3 = 64 3 . Simplifying both
solve x 3 = 64, raise both sides to the __
3
sides gives x = 4. Use this method to solve each equation. Check your answer.
1 x3 2
94. y 5 = 32 2
95. 27x 3 = 729 3
96. 1 = _
8
97. Geometry The formula for the surface area of a sphere S in terms of its volume V is
1
_
3
2
_
3 . What
S = (4π) (3V )
is the surface area of a sphere that has a volume of 36π cm ?
Leave the symbol π in your answer. What do you notice? 36π cm 2; both volume and
3
surface area are described by 36π (although the units are different).
Journal
Have students write the steps they
5
_
would use to simplify 729 6 .
SPIRAL REVIEW
Solve each equation. (Lesson 2-6)
98. ⎪x + 6⎥ = 2 -8, -4
100. ⎪2x - 1⎥ = 3 -1, 2
99. ⎪5x + 5⎥ = 0 -1
Have students write a quiz on
rational exponents. The quiz should
include five problems about simplifying expressions of various types. The
Solve each inequality and graph the solutions. (Lesson 3-4)
1x + 3
101. 3n + 5 < 14
102. 4 ≤ _
103. 7 ≥ 2y + 11
2
Give the domain and range of each relation. Tell whether the relation is a function.
Explain. (Lesson 4-2)
104. {(2, 3), (2, 4), (2, 5), (2, 6)}
105. {(-2, 0), (-1, 1), (0, 2), (1, 3)}
106.
107.
x
y
5
2
7
2
9
Ó
Simplify each expression.
ä
1. 16 4
1
_
Ó
2
11
7-5
{
Ó
{
Ó
2
7- 5 Rational Exponents
_______________________________________
__________________
__________________
Problem Solving
7-5 PROBLEM SOLVING
LESSON
7-5
Rational Exponents
1. For a pendulum with a length of L meters,
the time in seconds that it takes the
pendulum to swing back and forth is
2. The Beaufort Scale is used to measure
with Beaufort number B, the formula
1
Challenge
LESSON
7-5
7-5 CHALLENGE
6s
4. 64
START
7/18/09 4:53:59 PM
1
16 4
51.3 mi/h
1
42
−
1
27 3
1
2
05
1
2
13 − 8 3
216 3
1
3
83 + 92
4. At a factory that makes cylindrical cans,
1
V 2
the formula r = is used to find the
12 3
A2 .
by the formula V =
Find the volume
of a cube whose faces each have an area
of 64 in2.
1
1
36 2 − 216 3
radius of a can with volume V. What is
the radius of a can whose volume is
192 cm3?
1
4 cm
512 in3
1
_
- 27 3
9
729
7
_
6
128
5. In an experiment, the approximate population P of a
bacteria colony is given by
Keep Growing!
Find a path from start to finish in the maze below. Each box that you
pass through must have a value that is greater than or equal to the
value in the previous box. You may only move horizontally or
vertically to go from one box to the next.
3
3. Given a cube whose faces each have
area A, the volume of the cube is given
3.
3
_
81 2
2
Name ________________________________________ Date __________________ Class__________________
v = 1.9B 2 may be used to estimate the
tornado’s wind speed in miles per hour.
Estimate the wind speed of a tornado
with Beaufort number 9.
approximately 2L2 . About how long
A1NL11S_c07_0488_0493.indd does
493it take a pendulum that is 9 meters
long to swing back and forth?
493
2.
1
_
144 2
1
2
1
25 2 − 32 5
1
3
1
92 − 83
15
64 6
3
2
3
2
12 − 9 2
125 3
814
1000 3
1
1
2
100 2 + 27 3
1
1
16 2 − 16 4
3
32 5 + 0 4
5
_
P = 15t 3 , where t is the number of days since the start
of the experiment. Find the
population of the colony on
the 8th day. 480
3
Given an animal’s body mass m, in grams, the formula B = 1.8m 4
may be used to estimate the mass B, in grams, of the animal’s brain.
The table shows the body mass of several birds. Use the table for
questions 5–7. Select the best answer.
5. Which is the best estimate for the brain
mass of a macaw?
A 9g
C 125 g
B 45 g
D
6. How much larger is the brain mass of a
barn owl compared to the brain mass of a
cockatiel?
F 189 g
225 g
G
7. An animal has a body mass given by the
expression x 4 . Which expression can be
used to estimate the animal’s brain
mass?
A
B = 1.8x3
340.2 g
2
1
1
32 5
27 3
1210 + 12
1
3
16 4
3
1
16 4 − 12
1
102410
1
1
Simplify. All variables represent
nonnegative numbers.
9 2 − 30
1
4
814 + 49 2
243 5
1
1
1
2
16 4 + 32 5
3
100 2
625 4
128 7
1
1
144 2 − 812
1
64 3
5
6. √
x 10z 5
Body Mass (g)
Cockatiel
81
Guam Rail
256
D B = 1.8x
1
49 2 + 0 2
J 1215 g
Bird
C B = 1.8x12
1
4
92 + 34
H 388.8 g
Typical Body Masses of Birds
3
B B = 1.8x 4
2
512 9
Macaw
625
Barn Owl
1296
1
3
125 3 − 20
1
625 4
4
1
3
2
814 − 32 5
256 2
16 4 + 4 2
64 3
32 5 + 100 2
2
243 5
2
125 3
3
128 7
1
243 5
5
64 6
2
− 125 3
3
1
1
7.
(
)
1 43
_
a 4b 4
x 2z
√
b3
a 16b 2
Also available on transparency
Sources:
http://www.beyondveg.com/billings-t/companat/comp-anat-appx2.shtml
http://www.sandiegozoo.org/animalbytes/index.html
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FINISH
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Lesson 7-5
493
SECTION 7A
SECTION
7A
Exponents
I See the Light! The speed of light is the product of its
frequency f and its wavelength w. In air, the speed of light is
3 × 10 8
3 × 10 8 m/s.
8 ) -1
(
Organizer
then solve this equation for frequency. Write this equation as
an equation with w raised to a negative exponent.
Objective: Assess students’
GI
ability to apply concepts and skills
in Lessons 7-1 through 7-5 in a
real-world format.
<D
@<I
_; f =
3 × 10 w
w
1. Write an equation for the relationship described above, and
1. f =
2. Wavelengths of visible light range from 400 to 700 nanometers
(10 -9 meters). Use a graphing calculator and the relationship you found
in Problem 1 to graph frequency as a function of wavelength. Sketch the
graph with the axes clearly labeled. Describe your graph.
Online Edition
3. The speed of light in water is __34 of its speed in air. Find the speed of light
in water. 2.25 × 10 8 m/s
4. When light enters water, some colors bend more than others. How
much the light bends depends on its wavelength. This is what creates
a rainbow. The frequency of green light is about 5.9 × 10 14 cycles per
second. Find the wavelength of green light in water. about 3.81 × 10 -7 m
Resources
Algebra I Assessments
5. When light enters water,
www.mathtekstoolkit.org
Problem
Text Reference
1
Lesson 7-1
2
Lesson 7-2
3
Lesson 7-3
4
Lesson 7-3
5
Lesson 7-4
colors with shorter
wavelengths bend more
than colors with longer
wavelengths. Violet light
has a frequency of
7.5 × 10 14 cycles per
second, and red light has
a frequency of 4.6 × 10 14
cycles per second. Which
of these colors of light
will bend more when
it enters water? Justify
2.
Violet light will bend more.
Students can justify by using
the inverse relationship, by
using the graph, or by finding the wavelength for each
color and comparing them.


As the wavelength increases, the
frequency decreases.
494
Chapter 7 Exponents and Polynomials
5. How does wavelength change as
frequency changes? As one increases, the
other decreases.
INTERVENTION
Scaffolding Questions
A1NL11S_c07_0494.indd 494
1. What does it mean to “solve for
frequency”? Isolate the variable that
represents frequency.
2. What are a reasonable domain and
range? Possible answer: D: 380 to 760;
R: 3 × 1014 to 9 × 1014
3. How do you enter numbers in
scientific notation into a calculator?
Possible answer: 3 × 1014 is entered as
3
KEYWORD: MA7 Resources
494
Chapter 7
14 or 3
14.
4. What information do you need for this
problem? speed of light in water Where
can you find this information? answer to
Problem 3
6/25/09 9:15:23 AM
Extension
When you choose an FM radio station,
you are choosing the frequency in MHz, or
millions of waves per second (90 MHz =
90,000,000 waves per second). Find the
w = (3 × 108) f -1, FM 93.7 has an
approximate wavelength of 3.2 m.
SECTION 7A
SECTION
Quiz for Lessons 7-1 Through 7-5
7A
7-1 Integer Exponents
Evaluate each expression for the given value(s) of the variable(s).
_
_
2. n -3 for n = -5 - 1
1. t -6 for t = 2 1
Simplify.
5
_
4. 5k -3
k
3. r 0 s -2 for r = 8 and s = 10
125
64
x4
5. _
y -6
3
6. 8f -4 g 0
x 4y 6
8. Measurement Metric units can be written
in terms of a base unit. The table shows
some of these equivalencies. Simplify each
expression.
_8
f
a -3
7. _
b -2
4
1
_
100
b
_
Objective: Assess students’
a3
mastery of concepts and skills in
Lessons 7-1 through 7-5.
2
Selected Metric Prefixes
Milli-
Centi-
Deci-
10 -3
10 -2
10 -1
Deka- Hecto10 1
Organizer
Kilo-
10 2
10 3
Resources
7-2 Powers of 10 and Scientific Notation
Assessment Resources
10. Write 0.0000001 as a power of 10. 10 -7
12. Find the value of 82.1 × 10 4. 821,000
9. Find the value of 10 4. 10,000
11. Write 100,000,000,000 as a power of 10. 10 11
Section 7A Quiz
13. Measurement The lead in a mechanical pencil has a diameter of 0.5 mm. Write
this number in scientific notation. 5 × 10 -1
Test & Practice Generator
7-3 Multiplication Properties of Exponents
Simplify.
15. 3 5 · 3 -3 3 2 , or 9
14. 2 2 · 2 5 2 7
INTERVENTION
1
17. a 3 · a -6 · a -2 _
16. p 4 · p 5 p 9
a5
18. Biology A swarm of locusts was estimated to contain 2.8 × 10 10 individual insects.
If each locust weighs about 2.5 grams, how much did this entire swarm weigh? Write
Resources
Intervention and
Enrichment Worksheets
Simplify.
19.
(3x 4 )3 27x 12
20.
(m 3 n 2 )5 m 15 n 10
21.
(-4d 7 )2 16d 14
22.
(cd 6 )3 · (c 5 d 2 )2
c 13 d 22
7-4 Division Properties of Exponents
Simplify.
69
23. _
36
67
12a 5 4a 3
24. _
3a 2
25.
27
() _
125
3
_
5
3
26.
( )
4p 3
_
2pq 4
2
4p
_
4
q8
Simplify each quotient and write the answer in scientific notation.
27.
(8 × 10 9 ) ÷ (2 × 10 6 )
4 × 10
(3.5 × 10 5 ) ÷ (7 × 10 8 )
28.
5 × 10
3
-4
29.
(1 × 10 4 ) ÷ (4 × 10 4 )
2.5 × 10
-1
7-5 Rational Exponents
Simplify each expression. All variables represent nonnegative numbers.
1
_
1
_
30. 81 2 9
8 y4
34. √x
x 4y 2
3
_
31. 125 3 5
32. 4 2 8
3
35. √r 9
6
36. √z12
___
___
r3
2
_
33. 0 9 0
z2
______
37. √p 3 q12 pq 4
3
NO
A1NL11S_c07_0495.indd 495
Intervention
TO
Worksheets
103 = 1000
495
YES
Diagnose and Prescribe
INTERVENE
1
8. 10-3 = _, or 0.001;
1000
1
10-2 = _, or 0.01;
100
1
10-1 = _, or 0.1;
10
101 = 10; 102 = 100;
ENRICH
7/20/09 5:03:23 PM
GO ON? Intervention, Section 7A
CD-ROM
Lesson 7-1
7-1 Intervention
Activity 7-1
Lesson 7-2
7-2 Intervention
Activity 7-2
Lesson 7-3
7-3 Intervention
Activity 7-3
Lesson 7-4
7-4 Intervention
Activity 7-4
Lesson 7-5
7-5 Intervention
Activity 7-5
Online
Diagnose and
Prescribe Onlinew
Enrichment, Section 7A
Worksheets
CD-ROM
Online
495
SECTION
7B Polynomials
One-Minute Section Planner
Lesson
Lab Resources
Lesson 7-6 Polynomials
•
•
□
Optional
Classify polynomials and write polynomials in standard form.
Evaluate polynomial expressions.
SAT-10 ✔ NAEP
✔ ACT
✔ SAT
✔ SAT Subject Tests
□
□
□
Use algebra tiles to model polynomial addition and subtraction.
SAT-10 ✔ NAEP
ACT
SAT
SAT Subject Tests
□
□
□
□
Lesson 7-7 Adding and Subtracting Polynomials
•
□
✔ SAT-10 ✔ NAEP ✔ ACT
□
□
□
index cards, scissors, tape,
ruler (MK), 8.5-by-11 inch paper
□
7-7 Algebra Lab Model Polynomial Addition and Subtraction
•
□
Materials
SAT
□
Algebra Lab Activities
7-7 Lab Recording Sheet
Required
Technology Lab Activities
7-7 Technology Lab
Optional
Algebra Lab Activities
7-8 Lab Recording Sheet
Required
algebra tiles (MK)
books, pencils, slips of paper
SAT Subject Tests
7-8 Algebra Lab Model Polynomial Multiplication
• Use algebra tiles to model polynomial multiplication.
□ SAT-10 □ NAEP □ ACT □ SAT □ SAT Subject Tests
algebra tiles (MK)
Lesson 7-8 Multiplying Polynomials
• Multiply polynomials.
✔ NAEP □
✔ ACT
□ SAT-10 □
□ SAT
□ SAT Subject Tests
Lesson 7-9 Special Products of Binomials
• Find special products of binomials
✔ NAEP □
✔ ACT
□ SAT-10 □
□ SAT
□ SAT Subject Tests
Note: If NAEP is checked, the content is tested on either the Grade 8 or Grade 12 NAEP assessment.
496A
Chapter 7
MK = Manipulatives Kit
Math Background
TERMINOLOGY
Lesson 7-6
POLYNOMIAL OPERATIONS
Lessons 7-7 to 7-9
In order to discuss polynomials, we must agree on
terminology. The basic unit is the monomial. A
monomial is a product of a real number and one or
more variables with whole-number exponents. (The
real number is usually rational, particularly within the
scope of Algebra 1, but this is not a requirement.)
A polynomial is a sum of monomials. For example,
the polynomial 8x 4 - 3x - 1 may be written as 8x 4 +
(-3x) + (-1), which is the sum of the monomials
8x 4, -3x, and -1.
Adding and subtracting polynomials is fairly
straightforward because the process is nothing more
than combining like terms.
POLYNOMIALS
Lesson 7-6
Polynomials are in many ways analogous to counting
numbers. Because our number system is base 10, all
counting numbers can be written in expanded form
in terms of powers of 10. For example, consider the
expanded form of 653.
653 = 6 · 100 + 5 · 10 + 3 · 1
= 6 · 10 2 + 5 · 10 1 + 3 · 10 0
You can create a polynomial by replacing each of the
10s by a variable, such as x.
6 · 10 2 + 5 · 10 1 + 3 · 10 0
6 · x2 + 5 · x1 + 3 · x0
This polynomial is usually written in the more familiar
form 6x 2 + 5x + 3. For counting numbers, the only
permissible multipliers of the powers of 10 are the
digits 0 through 9, inclusive. For polynomials, any real
number can be a multiplier of the variable terms.
The goal of this analogy is not to suggest that there is
a correspondence between counting numbers and
polynomials but to demonstrate that diverse mathematical concepts sometimes share underlying structures. As such, it makes sense to pose some of the
same questions about polynomials that one might
pose about counting numbers. For example, can we
add, subtract, multiply, and divide polynomials? How?
(3x 2 + 7x + 5) + (2x + 6) = 3x 2 + 9x + 11
Polynomial multiplication can present greater difficulty for students, so it is essential to build gradually.
Multiplication of two monomials is a natural starting
point. You can use the Commutative and Associative
Properties to show that (4x)(2x) = 8x 2.
Use the Distributive Property when multiplying a
monomial and a binomial:
5x(2x + 3) = (5x)(2x) + (5x)(3) = 10x 2 + 15x
The Distributive Property is used repeatedly when
multiplying a binomial by a binomial:
(3x + 2)(7x + 4) = (3x)(7x + 4) + (2)(7x + 4)
= 21x 2 + 12x + 14x + 8
= 21x 2 + 26x + 8
In fact, the Distributive Property can be used to
multiply any two polynomials, regardless of the
number of terms. The product will have one term for
each product of a term from the first polynomial and a
term from the second polynomial. So, the product of a
binomial (2 terms) and a trinomial (3 terms) will have
2 · 3 = 6 terms before simplifying:
(x + 2)(x 2 + 6x + 8)
= (x + 2)(x 2) + (x + 2)(6x) + (x + 2)(8)
= (x)(x 2) + 2(x 2) + x(6x) + 2(6x) + x(8) + 2(8)
It is important to remember that this rule is true before
the product is simplified. Clearly, some of the terms
above are like terms and will be combined; the final
answer will have fewer than 6 terms. In general, the
product of a polynomial with m terms and a polynomial with n terms has mn terms before simplifying.
496B
7-6
Organizer
7-6
Block
Polynomials
__1 day
2
Objectives: Classify polynomials
and write polynomials in standard
form.
GI
Evaluate polynomial expressions.
@<I
<D
Evaluate polynomial
expressions.
Online Edition
Tutorial Videos
Countdown Week 16
Warm Up
Evaluate each expression for
the given value of x.
1. 2x + 3; x = 2
Who uses this?
Pyrotechnicians can use polynomials to plan
complex fireworks displays. (See Example 5.)
Objectives
Classify polynomials and
write polynomials in
standard form.
A monomial is a number, a variable, or a
product of numbers and variables with wholenumber exponents.
Vocabulary
monomial
degree of a monomial
polynomial
degree of a polynomial
standard form of a
polynomial
cubic
binomial
trinomial
Monomials
EXAMPLE
1
Find the degree of each monomial.
A -2a 2b 4
7.
2n7
4
B 4
69
2
6. y3
8.
Add the exponents of the variables: 2 + 4 = 6
The degree is 6.
2
4x 0
The degree is 0.
Identify the coefficient in each
term.
5. 4x3
2
-0.3x -2 4x - y _
x3
Finding the Degree of a Monomial
13
3. -4x - 2; x = -1
4. 7x2 + 2x; x = 3
0.5x
The degree of a monomial is the sum of the
exponents of the variables. A constant has
degree 0.
7
2. x2 + 4; x = -3
-7xy
5 x
Not Monomials
4
C 8y
1
1
-s4
There is no variable, but you can write 4 as 4x 0.
8y
The degree is 1.
-1
Also available on transparency
Find the degree of each monomial.
1a. 1.5k 2m 3
1b. 4x 1
The terms of an
expression are the
or subtracted. See
Lesson 1-7.
Q: What happened to the quadratic
polynomial when he fell asleep on
the beach?
A variable written without an exponent has exponent 1.
1c. 2c 3 3
A polynomial is a monomial or a sum or difference of monomials. The degree
of a polynomial is the degree of the term with the greatest degree.
EXAMPLE
2
Finding the Degree of a Polynomial
Find the degree of each polynomial.
A 4x - 18x 5
A: He got second-degree burns.
4x: degree 1
-18x 5: degree 5
Find the degree of each term.
The degree of the polynomial is the greatest degree, 5.
496
Chapter 7 Exponents and Polynomials
1 Introduce
A1NL11S_c07_0496-0501.indd
e x p l o r496
at i o n
7-6
Motivate
Polynomials
1. The table shows examples of expressions that are and are not
polynomials. What are some characteristics that the polynomials
have in common? Describe some ways in which the polynomials
are different from the expressions that are not polynomials.
Polynomials
Not Polynomials
4x 2 ⫺ 5x ⫹ 1
3x
0.6x ⫺1 ⫹ 7
8
3y ⫺ 0.5y
4
__
⫺9x 2
1
⫺2z 3 ⫹ 3z 2 ⫹ __
z
1.7xy
3xy
4
x2
3
⫺5
⫹ 3x y
2
⫺8
1 ⫹2
___
2
4x 2y ⫺ x 3y 2
xy
2. What do you notice about the exponents in the polynomials?
3. Do you think 16x ⫹ 2xy ⫹ 8y
2
KEYWORD: MA7 Resources
⫺2
Chapter 7
n: 2
p: 1
Ask students to identify the greatest exponent. 3
is a polynomial? Why or why not?
THINK AND DISCUSS
4. Show your own examples of expressions that are and are
not polynomials.
5. Describe how you can tell whether an expression is a polynomial.
496
Then have students identify the exponent of each
variable.
m: 3
3z ⫹ __
1
1 z 5 ⫺ 2z 4 ⫹ 8z 2 ⫹ __
__
2
Write m3 + n2 + p + 5 on the board. Ask students how many terms are in the expression. 4
Explorations and answers are provided in
Alternate Openers: Explorations Transparencies.
6/25/09 9:21:24 AM
Find the degree of each polynomial.
""
B 0.5x y + 0.25xy + 0.75
0.5x 2y : degree 3
0.25xy : degree 2
0.75: degree 0
The degree of the polynomial is the greatest degree, 3.
Students often confuse the degree
of a polynomial with the number of
terms. Students may be less likely
to confuse the two if they equate
the word degree with an exponentrelated phrase such as “maximum
power.”
C 6x 4 + 9x 2 - x + 3
9x 2: degree 2
-x: degree 1
6x 4: degree 4
The degree of the polynomial is the greatest degree, 4.
3: degree 0
Find the degree of each polynomial.
2a. 5x - 6 1
2b. x 3y 2 + x 2y 3 - x 4 + 2 5
The terms of a polynomial may be written in any order. However, polynomials
that contain only one variable are usually written in standard form.
The standard form of a polynomial that contains one variable is written with
the terms in order from greatest degree to least degree. When written in standard
form, the coefficient of the first term is called the leading coefficient .
EXAMPLE
3
Find the degree of each
monomial.
Write each polynomial in standard form. Then give the leading coefficient.
B. 7ed
C. 3
Find the degree of each term. Then arrange them in descending order.
1
3
0
- 4x 3 - x 2 + 20x + 2
2
3
2
1
0
The standard form is -4x 3 - x 2 + 20x + 2. The leading coefficient is -4.
B y 3 + y 5 + 4y
Degree: 3
5
1
Example 3
3
1
Write each polynomial in standard form. Then give the leading
coefficient.
3a. 16 - 4x 2 + x 5 + 9x 3
3b. 18y 5 - 3y 8 + 14y
x + 9x - 4x + 16; 1
5
3
2
-3y 8
+ 18y + 14y ; -3
5
Some polynomials have special names based on their degree and the number of
terms they have.
Degree
Name
Terms
Name
0
Constant
1
Monomial
1
Linear
2
Binomial
2
3
Trinomial
3
Cubic
4 or more
Polynomial
4
Quartic
5
Quintic
6 or more
6th degree, 7th degree, and so on
497
4
Write each polynomial in
standard form. Then give the
A. 6x - 7x5 + 4x2 + 9
-7x5 + 4x2 + 6x + 9; -7
B. y2 + y6 - 3y
y6 + y2 - 3y; 1
INTERVENTION
Questioning Strategies
EX AM P LE
7- 6 Polynomials
1
• If a monomial has more than one
variable with an exponent, how do
you determine its degree?
• Why is the degree of a constant
always zero?
2 Teach
EX AM P LE
A1NL11S_c07_0496-0501.indd 497
2
6/25/09 9:21:35 AM
Guided Instruction
Discuss with students examples and nonexamples of monomials. Explain how to
find the degree of a monomial and the
degree of a polynomial. Be sure students
get plenty of exposure to the new vocabulary in this lesson—it will be used often
throughout the remainder of this chapter
and Chapter 8.
Find the degree of each
polynomial.
5
The standard form is y 5 + y 3 + 4y. The leading coefficient is 1.
5
0
y 5 + y 3 + 4y
{
⎧
⎨
⎩⎧
⎨
⎩
{
⎧
⎨
⎩
⎧
⎨
⎩
y 3 + y 5 + 4y
2
A. 11x7 + 3x3 7
1
1
B. _ w2z + _ z 4 - 5
3
2
Find the degree of each term. Then arrange them in descending order.
A variable written
without a coefficient
has a coefficient of 1.
7
Example 2
⎧
⎨
⎩
⎧
⎨
⎩
⎧
⎨
⎩
⎧
⎨
⎩
⎧
⎨
⎩
⎧
⎨
⎩
⎧
⎨
⎩
⎧
⎨
⎩
20x - 4x 3 + 2 - x 2
y = 1y
Example 1
A. 4p4q3
A 20x - 4x 3 + 2 - x 2
5
Writing Polynomials in Standard Form
Degree:
Ê,,",
,/
2
Through Kinesthetic Experience
Divide students into groups of five, and have
each write a monomial on an index card in
large print. They may write a constant or a
monomial with a variable. Tell them the variable must be x, but it can be raised to any
power between 1 and 4, and can have any
coefficient. Then call out classifications such
as quadratic trinomial and quartic binomial.
Have students in each group stand and
arrange themselves to form that polynomial,
holding their monomial in front of them for
the rest of the class to see.
• Do coefficients affect the degree of
the polynomial? Explain.
EX AM P LE
3
• When is the coefficient of the first
term also the leading coefficient of
the polynomial?
Lesson 7-6
497
EXAMPLE
4
Classifying Polynomials
Classify each polynomial according to its degree and number of terms.
A 5x - 6
Degree: 1
Example 4
A.
+ 4n
Degree: 2
Terms: 3
y 2 + y + 4 is a quadratic trinomial.
C 6x 7 + 9x 2 - x + 3
cubic binomial
Degree: 7
B. 4y6 - 5y3 + 2y - 9
6th-degree polynomial
C. -2x
5x - 6 is a linear binomial.
B y +y+4
Classify each polynomial
according to its degree and
number of terms.
5n3
Terms: 2
2
Terms: 4
6x 7 + 9x 2 - x + 3 is a 7th-degree polynomial.
Classify each polynomial according to its degree and number of
terms.
4a. x 3 + x 2 - x + 2
4b. 6
4c. -3y 8 + 18y 5 + 14y
linear monomial
cubic polynomial constant monomial 8th-degree trinomial
Example 5
EXAMPLE
A tourist accidentally drops her
lip balm off the Golden Gate
Bridge. The bridge is 220 feet
from the water of the bay. The
height of the lip balm is given
by the polynomial -16t 2 + 220,
where t is time in seconds. How
far above the water will the lip
balm be after 3 seconds? 76 ft
5
Physics Application
A firework is launched from a platform 6 feet
above the ground at a speed of 200 feet per
second. The firework has a 5-second fuse.
The height of the firework in feet is given
by the polynomial -16t 2 + 200t + 6, where
t is the time in seconds. How high will the
firework be when it explodes?
¶
Substitute the time for t to find the
firework’s height.
-16t 2 + 200t + 6
-16(5)2 + 200 (5) + 6
-16(25) + 200 (5) + 6
-400 + 1000 + 6
INTERVENTION
ÈÊvÌ
The time is
5 seconds.
Evaluate the polynomial by using the
order of operations.
606
Questioning Strategies
When the firework explodes, it will be 606 feet above the ground.
EX A M P L E
4
5. What if…? Another firework with a 5-second fuse is launched
from the same platform at a speed of 400 feet per second. Its
height is given by -16t 2 + 400t + 6. How high will this firework
be when it explodes? 1606 ft
• Why will a constant polynomial
always be a monomial?
• How does writing a polynomial in
polynomial?
EX A M P L E
THINK AND DISCUSS
5
• What step in the order of operations will never be used when
evaluating a polynomial in standard
form? Why?
1. Explain why each expression is not a polynomial: 2x 2 + 3x -3; 1 - _ab_ .
2. GET ORGANIZED Copy and
complete the graphic organizer.
In each oval, write an example
of the given type of polynomial.
*Þ>Ã
>Ã
>Ã
498
Chapter 7 Exponents and Polynomials
3 Close
Summarize
A1NL11S_c07_0496-0501.indd 498
Have students give examples of
monomials, binomials, trinomials, and
polynomials of fourth and fifth degree.
Write their examples on the board as they
say them. For each, have students state
whether the polynomial is in standard
form, and if not, rewrite it in standard form.
and INTERVENTION
Diagnose Before the Lesson
7-6 Warm Up, TE p. 496
Monitor During the Lesson
Check It Out! Exercises, SE pp. 496–498
Questioning Strategies, TE pp. 497–498
Assess After the Lesson
7-6 Lesson Quiz, TE p. 501
Alternative Assessment, TE p. 501
498
Chapter 7
/À>Ã
1. Possible answer: 2x2 + 3x-3 contains
an expression with a negative
a
exponent. 1 - __
contains a variable
b
within a denominator.
2. See p. A7.
6/25/09 9:21:38 AM
7-6
Exercises
7-6 Exercises
KEYWORD: MA11 7-6
KEYWORD: MA7 Parent
GUIDED PRACTICE
Assignment Guide
Vocabulary Match each polynomial on the left with its classification on the right.
1. 2x 3 + 6 d
Assign Guided Practice exercises
as necessary.
a. quartic polynomial
2. 3x 3 + 4x 2 - 7 c
3. 5x - 2x + 3x - 6 a
c. cubic trinomial
2
4
If you finished Examples 1–3
Basic 27–49
Average 27–49
d. cubic binomial
SEE EXAMPLE
1
SEE EXAMPLE
Find the degree of each monomial.
5. -7xy 2 3
4. 10 6 0
p. 496
2
9. 0.75a 2b - 2a 3b 5 8
11. r + r - 5 3
3
SEE EXAMPLE
3
12. a + a - 2a 3
2
3
15. 9a - 8a
2
8
b 2 - 2b 2+ 5; 1
g 2 + 5g - 7; 1
5
p. 498
13. 3k 4 + k 3 - 2k 2 + k 4
16. 5s - 3s + 3 - s
If you finished Examples 1–5
Basic 27–78, 81–89
Average 27–79, 81–89
Homework Quick Check
7
-s 7 +
5s 24- 3s 3+ 3; -1
2
19. 3c + 5c + 5c - 4
Quickly check key concepts.
Exercises: 27, 32, 34, 42, 52, 58, 60
5c 4 + 5c 3 + 3c 2 - 4; 5
Classify each polynomial according to its degree and number of terms.
21. x- 7
trinomial
2
3
linear binomial
2
3
quartic polynomial
cubic binomial
23. q + 6 - q + 3q 4
SEE EXAMPLE
10. 15y - 84y 3 + 100 - 3y 2 3
2
18. 5g - 7 + g
3x 2 + 2x - 1; 3
20. x 2 + 2x + 3
p. 498
9
-8a 9 + 9a 28; -8
17. 2x + 3x - 1
SEE EXAMPLE 4
2
Write each polynomial in standard form. Then give the leading coefficient.
14. -2b + 5 + b
p. 497
7. 2 0
Find the degree of each polynomial.
8. x 2 - 2x + 1 2
p. 496
6. 0.4n 8 8
24. 5k + 7k
26. Geometry The surface area of a cone
is approximated by the polynomial
3.14r 2 + 3.14r, where r is the radius
and is the slant height. Find the
approximate surface area of
this cone. 301.44 cm2
22. 8 + k + 5k 4
quartic trinomial
3
2
25. 2a + 4a - a 4
quartic trinomial
->ÌÊi}ÌÊÊ£äÊV
ÈÊV
PRACTICE AND PROBLEM SOLVING
Independent Practice
For
See
Exercises Example
27–34
35–40
41–49
50–57
58
1
2
3
4
5
Find the degree of each monomial.
27. 3y 4 4
28. 6k 1
29. 2a 3b 2c 6
30. 325 0
31. 2y 4z 3 7
32. 9m 5 5
33. p 1
34. 5 0
Find the degree of each polynomial.
35. a 2 + a 4 - 6a 4
38. -5f 4 + 2f 6 + 10f
8
8
36. 3 2b - 5 1
37. 3.5y 2 - 4.1y - 6 2
39. 4n 3 - 2n 3
40. 4r 3 + 4r 6 6
Extra Practice
Skills Practice p. S17
Application Practice p. S34
Write each polynomial in standard form. Then give the leading coefficient.
41. 2.5 + 4.9t 3 - 4t 2 + t
42. 8a - 10a 2 + 2
43. x 7 - x + x 3 - x 5 + x 10
44. -m + 7 - 3m
45. 3x + 5x -2 4 + 5x
46. 2-2n + 1 - n 2
47. 4d + 3d - d + 5
48. 3s 2 + 12s + 6
49. 4x - x - x + 1
4.9t 3 - 4t 2 + t + 2.5;
4.9
2
7; 3-3
-3m - m +
2
2
-10a22 + 8a + 2; -10
3
5x 3
-d 3 + 3d 2 + 4d + 5; -1
x 10 + x 7 - x 5 + x 3 - x ; 1
+ 3x3 + 5x - 4; 5 -n - 22n +5 1; -1
3
12s 3 + 3s 2 + 6; 12
-x 5 - x 3 + 4x 2 + 1; -1
7- 6 Polynomials
499
Teacher to Teacher
For many students it is easier
to identify the degree of each
term in a polynomial when
each term is written with the
variables and their exponents.
For example:
A1NL11S_c07_0496-0501.indd 499
5x2y + 25xy + 75
= 5x2y1 + 25x1y1 + 75x0y0
5x2y1
degree 2 + 1 = 3
⎯⎯
25x1y1
degree 1 + 1 = 2
⎯⎯
75x0y0
degree 0 + 0 = 0
⎯⎯
The highest degree is 3, so the
polynomial 5x2y + 25xy + 75
has degree 3.
6/25/09 9:21:44 AM
Arlane Frederick
Buffalo, NY
KEYWORD: MA7 Resources
Lesson 7-6
499
Exercise 74 involves
expressing the area
and perimeter of a
rectangle as polynomials. This exercise prepares students for the MultiStep Test Prep on page 528.
Classify each polynomial according to its degree and number of terms.
Transportation
Kinesthetic Have students
make (or attempt to make)
the boxes for parts a, b,
and c of Exercise 63. Give students
scissors and tape and make sure
they have a ruler and 8.5-by-11-inch
paper (standard size of notebook or
copy paper). The ruler can be found
in the Manipulatives Kit (MK).
78. Time (s)
86
3
81
4
44
52. 3.5x 3 - 4.1x - 6 53. 4g + 2g 2 - 3
54. 2x - 6x
55. 6 - s - 3s
56. c 2 + 7 - 2c 3
linear3 monomial
4
quartic trinomial
cubic trinomial
57. -y 2
cubic trinomial
Tell whether each statement is sometimes, always, or never true.
Hybrid III is the crash
test dummy used by the
Insurance Institute for
Highway Safety. During
a crash test, sensors
neck, chest, legs, and
feet measure and record
forces. Engineers study
this data to help design
safer cars.
59
2
51. 6k
constant
monomial
2
58. Transportation The polynomial 3.675v + 0.096v 2 is used by transportation
officials to estimate the stopping distance in feet for a car whose speed is v miles
per hour on flat, dry pavement. What is the stopping distance for a car traveling at
30 miles per hour? 196.65 ft
Height (ft)
1
50. 12
59. A monomial is a polynomial. always
60. A trinomial is a 3rd-degree polynomial. sometimes
61. A binomial is a trinomial. never
62. A polynomial has two or more terms. sometimes
8.5 in.
63. Geometry A piece of 8.5-by-11-inch cardboard has
identical squares cut from its corners. It is then folded
into a box with no lid. The volume of the box in cubic
inches is 4c 3 - 39c 2 + 93.5c, where c is the side length of
the missing squares in inches.
a. What is the volume of the box if c = 1 in.? 58.5 in3
b. What is the volume of the box if c = 1.5 in.? 66
c. What is the volume of the box if c = 4.25 in.? 0
11 in.
in3
c
sense? Explain why or why not. Yes; the width of the cardboard is
The rocket will be highest after 2 s.
8.5 in., so 4.25 in. cuts will meet, leaving nothing to fold up.
First identify the
Copy and complete the table by evaluating each polynomial for the given values of x.
degree of each term.
From left to right,
Polynomial
x = -2
x=0
x=5
the degrees are
)
(
)
5x - 6
5 -2 - 6 = -16
5(0 - 6 = -6
64.
3, 0, 2, 4,
19
5
3
and 1.
x
+
x
+
4x
65.
-48
0
3270
Arrange the
-10x 2
66.
-40
-250
0
terms in
order of decreasing
degree, and move Give one example of each type of polynomial. Possible answers given.
the plus or minus 67. quadratic trinomial
68. linear binomial 5x - 2 69. constant monomial 5
x 2 + 3x - 6
3
70. cubic monomial 6x
71. quintic binomial x 5 - 3 72. 12th-degree trinomial
term with it:
2x 12 - x + 15
-2x 4 + 4x 3 +
73.
Write
It
Explain
the
steps
you
would
follow
to
write
the
polynomial
5x 2 - x - 3.
4x 3 - 3 + 5x 2 - 2x 4 - x in standard form.
c
74. This problem will prepare you for the Multi-Step Test Prep on page 528.
a. The perimeter of the rectangle shown is 12x + 6. What is
the degree of this polynomial? 1
ÓÝÊÊÎ
b. The area of the rectangle is 8x 2 + 12x. What is the degree
of this polynomial?
2
7-6 PRACTICE A
{Ý
7-6 PRACTICE C
Practice B
LESSON
7-6
Polynomials
7-6 PRACTICE B
500
Chapter 7 Exponents and Polynomials
Find the degree and number of terms of each polynomial.
3
2. 7y 10y 2
1. 14h + 2h + 10
3
3
3. 2a 2 5a + 34 6a 4
2
2
4
4
LESSON
7-6
Write each polynomial in standard form. Then, give the leading
coefficient.
4. 3x 2 2 + 4x 8 x
4x8 + 3x2 x 2
4
5. 7 50j + 3j 3 4j 2
3j3 4j 2 50j + 7
3
6. 6k + 5k 4 4k 3 + 3k 2
5k4 4k3 + 3k2 + 6k
5
Understanding Vocabulary
There is a great deal of introductory vocabulary related to polynomials.
The meaning of and relationships among the terms are shown below.
Classification Degree
Constant
0
Linear
1
2
Cubic
3
Quartic
4
Quintic
5
6th degree
6
7th degree
7
etc.
etc.
A1NL11S_c07_0496-0501.indd 500
Classify each polynomial by its degree and number of terms.
7. 5t 2 + 10
8. 8w 32 + 9w 4
quartic trinomial
9. b b 3 2b 2 + 5b 4
quartic
polynomial
10. 3m + 8 2m 3 for m = 1
1
12. 2w + w w 2 for w = 2
2
3
7
9
10
b. How high is the egg above the ground after 6 seconds?
2. Why does 5xy 3 have a degree of 4? (Hint: Look at the exponents on the variables.)
3. Classify 4n 3 + 6 by degree. cubic by number of terms. binomial
Use the polynomial 8g + 1 − 4g 2 to complete the following.
4. Write the polynomial in standard form. −4g
2
+ 8g + 1
5. What is the leading coefficient? −4
6. What is the degree of the polynomial? 2
7. Classify the polynomial by number of terms. trinomial
500
8. Classify the polynomial by degree. quadratic
Chapter 7
7-6 RETEACH
The degree of the monomial is the sum of the exponents in the monomial.
Find the degree of 8x 2 y 3 .
Find the degree of −4a 6 b.
8x 2 y 3 The exponents are 2 and 3.
−4a 6 b The exponents are 6 and 1.
The degree of the monomial
is 2 + 3 = 5.
The degree of the monomial
is 6 + 1 = 7.
The degree of the polynomial is the degree of the term with the greatest degree.
Find the degree of 2x 4 y 3 + 9x 5 .
Find the degree of 4ab + 9a 3 .
2
x 4 y
3 + 9
x5
N
5
7
4
ab + 9
a3
N
N
2
3
Degree of the
polynomial is 7.
Degree of the
polynomial is 3.
5
x + 6
x3 + N
4 − 2
x4
N
N
N
4
1
3
0
Find the degree of each term.
2x 4 + 6x 3 + 5x + 4
Write the terms in order of degree.
Find the degree of each monomial.
Complete each of the following.
The exponents on the variables have a sum of 4.
135.6 m
Polynomials
1. How many terms are in a trinomial? 3 a monomial? 1
187.5 m
Review for Mastery
Write 5x + 6x 3 + 4 + 2x 4 in standard form.
13. An egg is thrown off the top of a building. Its height in meters above
the ground can be approximated by the polynomial 300 + 2t 4.9t 2,
where t is the time since it was thrown in seconds.
a. How high is the egg above the ground after 5 seconds?
7-6
A monomial is a number, a variable, or a product of numbers and variables with wholenumber exponents. A polynomial is a monomial or a sum or difference of monomials.
The standard form of a polynomial is written with the terms in order from the greatest
degree to the least degree. The coefficient of the first term is the leading coefficient.
Evaluate each polynomial for the given value.
11. 4y 5 6y + 8y 2 1 for y = 1
LESSON
1. 7m 3 n 5
8
2. 6xyz
3. 4x 2 y 2
3
4
Find the degree of each polynomial.
4. x 5 + x 5 y
6
5. 4x 2 y 3 + y 4 + 7
5
6. x 2 + xy + y
2
Write each polynomial in standard form. Then give the leading
coefficient.
7. x 3 − 5x 4 − 6x 5
−6x 5 − 5x 4 + x 3
−6
8. 2x + 5x 2 − x 3
−x 3 + 5x 2 + 2x
−1
9. 8x + 7x 2 − 1
7x 2 + 8x −1
7
6/25/09 9:21:50 AM
75.
In Exercise 76, students who chose A
may have confused
degree with number of terms.
Students who chose D may think
that the coefficients are significant
when determining degree.
/////ERROR ANALYSIS///// Two students evaluated 4x - 3x 5 for x = -2. Which is
incorrect? Explain the error.
75. A is incorrect. The student incorrectly
multiplied -3 by -2 before evaluating
the power.
-!+",!+" .
1/ .
1000/
00/1
-!+",!+" .
1,!,+"
12/
11
76. Which polynomial has the highest degree?
3x 8 - 2 x 7 + x 6
5x - 100
In Exercise 77, students who chose
G probably got -3 for the value of
the first term.
25x 10 + 3x 5 - 15
134x 2
77. What is the value of -3x 3 + 4x 2 - 5x + 7 when x = -1?
3
13
9
80b. Yes; 0 < x < 1; raising a number
between 0 and 1 to a higher
power results in a lesser number.
So if x is between 0 and 1, the
binomial with the least degree
will have the greatest value.
19
78. Short Response A toy rocket is launched from the ground at 75 feet per
second. The polynomial -16t 2 + 75t gives the rocket’s height in feet after
t seconds. Make a table showing the rocket’s height after 1 second, 2 seconds,
3 seconds, and 4 seconds. At which of these times will the rocket be the highest?
Journal
CHALLENGE AND EXTEND
79. Medicine Doctors and nurses use growth charts and formulas to tell whether a
baby is developing normally. The polynomial 0.016m 3 - 0.390m 2 + 4.562m + 50.310
gives the average length in centimeters of a baby boy between 0 and 10 months of
age, where m is the baby’s age in months.
a. What is the average length of a 2-month-old baby boy? a 5-month-old baby boy?
79c. The first 3 terms
of the polynomial will
equal 0, so just look at b. What is the average length of a newborn (0-month-old) baby boy? 50.310 cm
c. How could you find the answer to part b without doing any calculations?
the constant.
80. Consider the binomials 4x 5 + x, 4x 4 + x, and 4x 3 + x.
a. Without calculating, which binomial has the greatest value for x = 5? 4x 5 + x
b. Are there any values of x for 4x 3 + x which will have the greatest value? Explain.
Have students explain how to classify a polynomial according to its
degree and number of terms.
Have students write a linear binomial
and a quadratic trinomial, each with
only x as a variable. Then have them
evaluate each polynomial for x = 3.
7-6
SPIRAL REVIEW
81. Jordan is allowed 90 minutes of screen time per day. Today, he has already used m
minutes. Write an expression for the remaining number of minutes Jordan has today.
(Lesson 1-1) 90 - m
83. incons.; no sol.
84. cons. and
dep.; inf. many
solutions
82. Pens cost \$0.50 each. Giselle bought p pens. Write an expression for the total cost of
Giselle’s pens. (Lesson 1-1) 0.50p
Classify each system. Give the number of solutions. (Lesson 6-4)
⎧ y = -4x + 5
83. ⎨
⎩ 4x + y = 2
85. cons. and
indep.; one sol.
⎧ 2x + 8y = 10
84. ⎨
⎩ 4y = -x + 5
Simplify. (Lesson 7-4)
4 7 4 3, or 64
86. _
44
_
x 6y 4 x 2
87. _
x 4y 9 y 5
( )
2v 4
_
vw 5
2
4v
_
( )
89.
w 10
_
7- 6 Polynomials
Name ________________________________________ Date __________________ Class__________________
LESSON
7-6
Problem Solving
7-6 PROBLEM SOLVING
Polynomials
1. The surface area of a cylinder is given
by the polynomial 2πr 2 + 2πrh. A cylinder
has a radius of 2 centimeters and a
height of 5 centimeters. Find the surface
area of the cylinder. Use 3.14
for π.
A1NL11S_c07_0496-0501.indd 501
2. A firework is launched from the ground
at a velocity of 180 feet per second. Its
height after t seconds is given by the
polynomial −16t 2 + 180t. Find the height
of the firework after 2 seconds and after
5 seconds.
87.92 square centimeters
2 s: 296 feet
5 s: 500 feet
3. In the United Kingdom, transportation
1 2
v +v
authorities use the polynomial
20
4. A piece of cardboard that measures 2
feet by 3 feet can be folded into a box
if notches are cut out of the corners.
The length of the side of the notch will
be the same as the height h of the
resulting box. The volume of the box is
given by 4h 3 − 10 h 2 + 6h. Find the
volume of the box for h = 0.25 and
h = 0.5.
for calculating the number of feet
needed to stop on dry pavement. In the
United States, many use the
polynomial 0.096v 2 . Both formulas are
based on speed v in miles per hour.
Calculate the stopping distances for a
car traveling 45 miles per hour in both
the U.S. and the UK.
h = 0.25: 0.9375 cubic feet
h = 0.5: 1 cubic foot
UK: 146.25 feet
US: 194.4 feet
The height of a rocket in meters t seconds after it is launched is
approximated by the polynomial 0.5at 2 + vt + h where a is always −9.8,
v is the initial velocity, and h is the initial height. Use this information
with the data in the chart for questions 5 – 7. Select the best answer.
5. A 300X was launched from a height of
10 meters. What was its height after 3
seconds?
A
715.9 m
B 745.3 m
Model Number
300X
Q99
4400i
C 755.5 m
D 760 m
6. Marie and Bob launched their rockets at
the same time from a platform 5 meters
above the ground. Marie launched the
4400i and Bob launched the Q99. How
much higher was Marie’s rocket after 2
seconds?
7-6
Challenge
7-6 CHALLENGE
Pick the Polynomial
Match each polynomial with the correct clue. Each polynomial can be
used only once. Not every polynomial will be used.
Use these polynomials for 1 – 7.
5
3
x +x +x
2x 4
3x + 3y + 3z
2xy + 5xz
4x 2 3x 5 + x
4 4
Use these polynomials for 8 – 14.
2 5
2
1. I am a monomial with degree 5.
Who am I?
7. The 4400i was launched from the ground
at the same time the Q99 was launched
from 175 meters above the ground. After
how many seconds were the rockets at
the same height?
H 140 meters
A 2s
C5 s
G
J 320 meters
B 4s
D 6s
3
2
x +x +x
x4 + x3 + x2
2x 2 3x 3 + 1
x3
x2 + y2 + z2 + w2
8. I am a linear expression. My
constant is 3. Who am I?
3xy4
x 3
2. I am a sum of monomials with
degree 8. Who am I?
9. I am a quartic trinomial. I have three
different variables. Who am I?
x3 y 3xyz + z
2x4 y4 + 3x 2 y5
3. I am a trinomial with degree 5.
Who am I?
10. I am a cubic binomial. Who am I?
4x2 3x 2 y
x5 + x3 + x
11. I am a quadratic polynomial. I have
no constants. Who am I?
x2 + y2 + z 2 + w2
2xy + 5xz
Initial Velocity
(m/s)
250
90
125
2
4x 3x y
x2 + x 3
3 + x + 4x 2
x 3 y 3xyz + z
x + 3
2x y + 3x y
x 5 y 3x 2 y xy
3xy 4
2x 2 y + 5xy 2
4xyz 2 + xyz
4. I am a binomial. Both of my terms
have degree 2. Who am I?
F 35 meters
70 meters
Name _______________________________________ Date __________________ Class__________________
LESSON
12. I am a cubic trinomial with one
variable. Who am I?
5. I am a monomial with degree 4.
Who am I?
x3 + x2 + x
2x4
13. I am a quadratic trinomial. When you put
me in standard form, my leading
coefficient is 4. Who am I?
6. I am a binomial with degree 4.
Who am I?
4xyz 2 + xyz
3 + x + 4x2
7. I am a trinomial. When you put me in
is 3. Who am I?
2. 25x2 - 3x4
14. I am a quartic trinomial. Who am I?
x4 + x3 + x2
7-49
Holt McDougal Algebra 1
7
4. 14 - x4 + 3x2 -x4 + 3x2 + 14;
-1
501
Classify each polynomial
according to its degree and
number of terms.
5. 18x2 - 12x + 5
6/25/09 9:21:57 AM
6. 2x4 - 1
quartic binomial
7. The polynomial
3.675v + 0.096v2 is used to
estimate the stopping distance
in feet for a car whose speed
is v miles per hour on flat,
dry pavement. What is the
stopping distance for a car
traveling at 70 miles per hour?
727.65 ft
Also available on transparency
4x2 3x5 + x
Lesson 7-6
Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.
5
4
3. 24g3 + 10 + 7g5 - g2
7g5 + 24g3 - g2 + 10;
2p -4 p 8
_
p3
16
6
1. 7a3b2 - 2a4 + 4b - 15
Write each polynomial in
standard form. Then give the
⎧ y = 3x + 2
85. ⎨
⎩ y = -5x - 6
88.
Find the degree of each
polynomial.
Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.
7-48
Holt McDougal Algebra 1
501
7-7
Organizer
and Subtraction
Use with Lesson 7-7
Pacing:
1
day
2
1
__
Block 4 day
You can use algebra tiles to model polynomial addition and subtraction.
Objective: Use algebra tiles to
Use with Lesson 7-7
subtraction.
KEY
GI
Materials: algebra tiles
@<I
<D
=1
= -1
Online Edition
KEYWORD: MA11 LAB7
=x
= x2
= -x
= -x 2
Algebra Tiles
Countdown Week 16
Activity 1
Use algebra tiles to find (2x 2 - x) + (x 2 + 3x - 1).
Resources
MODEL
Algebra Lab Activities
ALGEBRA
7-7 Lab Recording Sheet
Teach
Use tiles to represent all terms from
both expressions.
(2x 2 - x) + (x 2 + 3x - 1)
Rearrange tiles so that like tiles are
together. Like tiles are the same
size and shape.
(2x 2 + x 2) + (-x + 3x) - 1
Discuss
Compare like terms and like tiles. In
like terms, the same variables are
raised to the same powers, but the
coefficients may differ. In like tiles,
the size and shape are the same, but
the color of the tile (whether it is
positive or negative) may differ.
Remove any zero pairs.
In Activity 2, remind students to add
the opposite of every term in the
second expression.
The remaining tiles represent the
sum.
Encourage students to always
arrange their tiles and write their
answers in standard form. This will
help prevent careless errors.
3x 2 - x + x + 2x - 1
3x 2 + 2x - 1
Try This
Use algebra tiles to find each sum.
2.
(3x 2 + 2x + 5) + (x 2 - x - 4) 4x 2 + x + 1
3. (x - 3) + (2x - 2) 3x - 5
4.
(5x 2 - 3x - 6) + (x 2 + 3x + 6) 6x 2
5. -5x 2 + (2x 2 + 5x) -3x 2 + 5x
6.
(x 2 - x - 1) + (6x - 3) x 2 + 5x - 4
1.
502
(-2x 2 + 1) + (-x 2) -3x 2 + 1
Chapter 7 Exponents and Polynomials
A11NLS_c07_0502-0503.indd 502
KEYWORD: MA7 Resources
502
Chapter 7
12/10/09 8:48:46 PM
Close
Activity 2
Key Concept
Use algebra tiles to find (2x 2 + 6) - 4x 2.
MODEL
Polynomial addition is the same as
combining like terms. Polynomial
opposite.
ALGEBRA
Use tiles to represent
the terms in the first
expression.
2x 2 + 6
Assessment
Journal Have students explain
how to use algebra tiles to find
(3x2 - 4x + 1) - 5x.
To subtract 4x 2, you would remove 4 yellow x 2-tiles, but there are not
enough to do this. Remember that subtraction is the same as adding
the opposite, so rewrite (2x 2 + 6) - 4x 2 as (2x 2 + 6) + (-4x 2).
7. 3x2 + 4x
8. 2x2 - 4x - 7
2x 2 + 6 + (-4x 2)
9. 3x
10. 10x + 5
11. 5x2 + x
12. 2x2 - 6x - 4
Rearrange tiles so that
like tiles are together.
2x + (-4x )+ 6
2
13.
2
2x 2 + (-2x 2) + (-2x 2) + 6
Remove zero pairs.
The remaining tiles
represent the difference.
-2x 2 + 6
Try This
Use algebra tiles to find each difference.
(6x 2 + 4x) - 3x 2
8.
10. (8x + 5) - (-2x)
11.
7.
13.
(2x 2 + x - 7) - 5x
(x 2 + 2x) - (-4x 2 + x)
9. (3x + 6) - 6
12.
(3x 2 - 4) - (x 2 + 6x)
represents a zero pair. Use algebra tiles to model two other zero pairs.
14. When is it not necessary to “add the opposite” for polynomial subtraction using
algebra tiles? when you have enough tiles to actually remove them to model the subtraction
7- 7 Algebra Lab
A1NL11S_c07_0502-0503.indd 503
503
7/18/09 5:01:25 PM
7-7 Algebra Lab
503
7-7
Organizer
7-7
Block
__1 day
Polynomials
2
Who uses this?
subtract polynomials that model
profit. (See Example 4.)
Objective
polynomials.
Technology Lab
GI
In Technology Lab Activities
<D
@<I
Just as you can perform operations
on numbers, you can perform
operations on polynomials. To
combine like terms.
Online Edition
Tutorial Videos
Countdown Week 16
EXAMPLE
1
www.cartoonstock.com
polynomials.
A 15m 3 + 6m 2 + 2m 3
Warm Up
15m 3 + 6m 2 + 2m 3
15m 3 + 2m 3 + 6m 2
3
17m + 6m 2
Simplify each expression by
combining like terms.
1. 4x + 2x
6x
2. 3y + 7y
10y
3. 8p - 5p
3p
4. 5n + 6n2
not like terms
Simplify each expression.
5. 3(x + 4)
Identify like terms.
Rearrange terms so that like terms are together.
Combine like terms.
B 3x 2 + 5 - 7x 2 + 12
Like terms are
constants or terms
with the same
variable(s) raised to
the same power(s).
To review combining
like terms, see
Lesson 1-7.
3x 2 + 5 - 7x 2 + 12
2
3x - 7x 2 + 5 + 12
-4x 2 + 17
Identify like terms.
Rearrange terms so that like terms are together.
Combine like terms.
C 0.9y - 0.4y + 0.5x + y
5
5
5
5
0.9y 5 - 0.4y 5 + 0.5x 5 + y 5
0.9y 5 - 0.4y 5 + y 5 + 0.5x 5
1.5y 5 + 0.5x 5
3x + 12
6. -2(t + 3) -2t - 6
Identify like terms.
Rearrange terms so that like terms are together.
Combine like terms.
D 2x y - x y - x y
2
2
2
2x 2y - x 2y - x 2y
0
7. -1( - 4x - 6)
-x2 + 4x + 6
x2
Also available on transparency
All terms are like terms.
Combine.
1a. 2s 2 + 3s 2 + s 5s 2 + s
1b. 4z 4 - 8 + 16z 4 + 2 20z 4 - 6
8
8
8
8
8 8
1c. 2x + 7y - x - y x + 6y 1d. 9b 3c 2 + 5b 3c 2 - 13b 3c 2 b 3c 2
Polynomials can be added in either vertical or horizontal form.
In vertical form, align the like terms
Teacher: What is b + b?
Shakespeare: Is it 2b or not 2b?
(5x 2 + 4x + 1) + (2x 2 + 5x + 2)
= (5x 2 + 2x 2) + (4x + 5x) + (1 + 2)
5x 2 + 4x + 1
+ 2x 2 + 5x + 2
7x 2 + 9x + 3
504
In horizontal form, use the Associative
and Commutative Properties to
regroup and combine like terms:
= 7x 2 + 9x + 3
Chapter 7 Exponents and Polynomials
1 Introduce
A1NL11S_c07_0504-0509.indd
e x p l o r504
at i o n
7-7
Polynomials
An ecologist is studying frogs and toads in a wetlands habitat.
She ﬁnds that the polynomial 3x 2 ⫹ x models the frog population
and that the polynomial 7x 2 ⫺ x models the toad population. In
both cases, x represents the number of months.
1. Complete the table. For each month, evaluate the two
polynomials to find the population of frogs and toads. Then
add these values to find the total population.
Population of Population of
Month, x Frogs, 3x 2 ⫹ x Toads, 7x 2 ⫺ x
1
4
6
2
3
4
5
6
Total
Population
10
2. Look for a pattern in the last column. Write a simplified
polynomial that gives the total population in month x.
KEYWORD: MA7 Resources
THINK AND DISCUSS
3. Show how to find the sum of the polynomials 3x 2 ⫹ x and
7x 2 ⫺ x.
4. Describe how you could find the sum using the terms of the
two polynomials that are being added.
504
Chapter 7
Motivate
Display the following items: 4 books, 3 pencils,
2 books, and 5 pencils.
Ask students for a sensible way to group the
items. a group of 6 books and a group of
8 pencils
Explain to students that adding and subtracting
monomials is done in a similar way.
Explorations and answers are provided in
Alternate Openers: Explorations Transparencies.
6/25/09 9:19:47 AM
EXAMPLE
2
""
A (2x 2 - x) + (x 2 + 3x - 1)
(2x 2 - x) + (x 2 + 3x - 1)
(2x 2 + x 2) + (-x + 3x) + (-1)
3x 2 + 2x - 1
Students often think the terms xy2
and x2y are like terms. Write out
all factors to show students how
they are different: xy2 = xyy, and
x2y = xxy.
Identify like terms.
Group like terms together.
Combine like terms.
B (-2ab + b) + (2ab + a)
When you use
the Associative
and Commutative
Properties to
rearrange the terms,
each term must stay
with that term.
C
(-2ab + b) + (2ab + a)
(-2ab + 2ab) + b + a
Identify like terms.
0+b+a
b+a
Combine like terms.
Group like terms together.
(4b + 8b) + (3b + 6b - 7b + b)
(4b 5 + 8b) + (3b 5 + 6b - 7b 5 + b)
(4b 5 + 8b) + (-4b 5 + 7b)
5
5
4b + 8b
+ -4b 5 + 7b
5
+ 15b
0
15b
D
Simplify.
5
Example 1
Identify like terms.
Combine like terms in the second
polynomial.
Use the vertical method.
A. 12p3 + 11p2 + 8p3
20p3 + 11p2
Combine like terms.
B. 5x2 - 6 - 3x + 8 5x2 - 3x + 2
Simplify.
(20.2y + 6y + 5) + (1.7y - 8)
(20.2y 2 + 6y + 5) + (1.7y 2 - 8)
2
C. t2 + 2s2 - 4t2 - s2 -3t 2 + s2
2
20.2y 2 + 6y + 5
+ 1.7y 2 + 0y - 8
21.9y 2 + 6y - 3
D. 10m2n + 4m2n - 8m2n 6m2n
Identify like terms.
Use the vertical method.
Example 2
Write 0y as a placeholder in the
second polynomial.
A. (4m2 + 5) + (m2 - m + 6)
5m2 - m + 11
Combine like terms.
2. Add (5a 3 + 3a 2 - 6a + 12a 2) + (7a 3 - 10a).
12a 3 + 15a 2 - 16a
To subtract polynomials, remember that subtracting is the same as adding the
opposite. To find the opposite of a polynomial, you must write the opposite of
each term in the polynomial:
- (2x 3 - 3x + 7) = -2x 3 + 3x - 7
EXAMPLE
3
Ê,,",
,/
Subtracting Polynomials
B. (10xy + x) + (-3xy + y)
7xy + x + y
C. (6x2 - 4y) + (3x2 + 3y 8x2 - 2y) x2 - 3y
1
D. _ a2 + b + 2 +
2
3 2
_
a - 4b + 5 2a2 - 3b + 7
2
(
(
)
)
Subtract.
A
(2x 2 + 6) - (4x 2)
(2x 2 + 6) + (- 4x 2)
(2x 2 + 6) + (-4x 2)
(2x 2 - 4x 2) + 6
-2x 2 + 6
Rewrite subtraction as addition of the opposite.
Identify like terms.
Questioning Strategies
Group like terms together.
Combine like terms.
EX AM P LE
B (a 4 - 2a) - (3a 4 - 3a + 1)
(a 4 - 2a) + (- 3a 4 + 3a - 1) Rewrite subtraction as addition of the opposite.
(a 4 - 2a) + (- 3a 4 + 3a - 1) Identify like terms.
(a 4 - 3a 4) + (- 2a + 3a) - 1 Group like terms together.
-2a 4 + a - 1
Combine like terms.
7- 7 Adding and Subtracting Polynomials
6/25/09
Guided Instruction
Review like terms before beginning this
lesson. When rearranging terms, remind
students to pay close attention to the
in Example 2C, like terms in the second
polynomial were simplified before the
polynomials were added. Students will be
less likely to make mistakes when the individual polynomials are simplified first.
Visual Cues Draw different
marks around like terms so they
stand out.
Through Cooperative Learning
Prepare a bag with several small slips of
paper (at least four times the number of
students in class), each with one of the following: x, 2x, -3x, y, 4y, -6y. Have each
student randomly select three or four slips
of paper. Have students combine, and simplify if possible, their terms and write the
resulting polynomial on a piece of paper.
Then have students pair up to find the
sums and differences of their polynomials.
Have students pair up with as many others
as time allows.
1
• How do you identify like terms?
EX AM P LE
2
• When using a vertical format to
505
2 Teach
A1NL11S_c07_0504-0509.indd 505
INTERVENTION
Inclusion Remind
students that the
Commutative Property
of Addition states that you can
9:19:56
AM numbers in any order. The
states that you can group any of
the numbers together.
Lesson 7-7
505
Multiple Representations
Finding the opposite of a
polynomial can be thought
of as distributing -1.
Subtract.
C
(3x 2 - 2x + 8) - (x 2 - 4)
(3x 2 - 2x + 8) + (- x 2 + 4)
(3x 2 - 2x + 8) + (- x 2 + 4)
3x 2 - 2x + 8
+ -x 2 + 0x + 4
2x 2 - 2x + 12
(11z 3 - 2z) + (- z 3 + 5)
(11z 3 - 2z) + (- z 3 + 5)
Subtract.
-x3 + 4y
11z 3 - 2z + 0
+ -z 3 + 0z + 5
B. (7m4 - 2m2) (5m4 - 5m2 + 8)
2m4 + 3m2 - 8
10z - 2z + 5
3
C. (-10x2 - 3x + 7) - (x2 - 9)
-11x2 - 3x + 16
EXAMPLE
4
A farmer must add the areas of
two plots of land to determine
the amount of seed to plant. The
area of plot A can be represented
by 3x2 + 7x - 5, and the area of
plot B can be represented by
5x2 - 4x + 11. Write a polynomial that represents the total area
of both plots of land.
8x2 + 3x + 6
(- 0.03x 2 + 25x - 1500)
- (-0.02x 2 + 21x - 1700)
+ (+ 0.02x 2 - 21x + 1700)
Write 0 and 0z as placeholders.
Combine like terms.
Southern:
0.02x 2 21x 1700
Eastern:
0.03x2 25x 1500
Eastern plant profits
Southern plant profits
Write subtraction as addition of the opposite.
Combine like terms.
-0.05x 2 + 46x - 3200
3
THINK AND DISCUSS
• What is the first step in subtracting
polynomials?
1. Identify the like terms in the following list: -12x 2, -4.7y, __15 x 2y, y, 3xy 2,
-9x 2, 5x 2y, -12x
• Why are parentheses used when
subtracting polynomials?
2. Describe how to find the opposite of 9t 2 - 5t + 8.
3. GET ORGANIZED Copy and complete
the graphic organizer. In each box, write
an example that shows how to perform
the given operation.
4
• What is the purpose of aligning
polynomials?
506
*Þ>Ã
``}
-ÕLÌÀ>VÌ}
Chapter 7 Exponents and Polynomials
3 Close
A1NL11S_c07_0504-0509.indd 506
Have students list the steps for adding and
subtracting polynomials.
1. Rewrite subtraction as addition if
necessary.
2. Identify like terms.
3. Rearrange terms so that like terms
are together.
Chapter 7
Use the vertical method.
4. Use the information above to write a polynomial that
represents the total profits from both plants.
Questioning Strategies
506
Identify like terms.
-0.03x + 25x - 1500
INTERVENTION
5. Simplify if necessary.
Rewrite subtraction as addition of the opposite.
2
-0.01x + 4x + 200
4. Combine like terms.
Combine like terms.
Write a polynomial that represents the difference of the profits at the
eastern plant and the profits at the southern plant.
2
Summarize
Write 0x as a placeholder.
The profits of two different
manufacturing plants can
be modeled as shown,
where x is the number
of units produced
at each plant.
Example 4
EX A M P L E
Use the vertical method.
3. Subtract (2x 2 - 3x 2 + 1) - (x 2 + x + 1). -2x 2 - x
D. (9q2 - 3q) - (q2 - 5)
8q2 - 3q + 5
EX A M P L E
Identify like terms.
D (11z 3 - 2z) - (z 3 - 5)
Example 3
A. (x3 + 4y) - (2x3)
Rewrite subtraction as addition of the opposite.
and INTERVENTION
Diagnose Before the Lesson
7-7 Warm Up, TE p. 504
1. -12x2 and -9x2; -4.7y and
1
y; _ x2y and 5x2y
5
2. Take the opposite of each term:
-9t2 + 5t - 8.
3. See p. A7.
Monitor During the Lesson
Check It Out! Exercises, SE pp. 504–506
Questioning Strategies, TE pp. 505–506
Assess After the Lesson
7-7 Lesson Quiz, TE p. 509
Alternative Assessment, TE p. 509
6/25/09 9:19:59 AM
7-7
Exercises
7-7 Exercises
KEYWORD: MA11 7-7
KEYWORD: MA7 Parent
GUIDED PRACTICE
SEE EXAMPLE
1
Assignment Guide
p. 504
1. 7a 2 - 10a 2 + 9a
2. 13x 2 + 9y 2 - 6x 2
3. 0.07r 4 + 0.32r 3 + 0.19r 4
_
5. 5b 3c + b 3c - 3b 3c
6. -8m + 5 - 16 + 11m
-3a 2
+ 9a
7x + 9y
2
1 p3 + _
2 p3
4. _
11 p 3
4
3
2
p. 505
7.
9.
SEE EXAMPLE
3
p. 505
(5n 3 + 3n + 6) + (18n 3 + 9)
(-3x + 12) + (9x 2 + 2x - 18)
8.
10.
(6c 4 + 8c + 6) - (2c 4) 4c 4 + 8c + 6
12.
13. (2r + 5) - (5r - 6) -3r + 11
SEE EXAMPLE 4
p. 506
3
3m - 11
(3.7q 2 - 8q + 3.7) + (4.3q 2 - 2.9q + 1.6)
(9x 4 + x 3) + (2x 4 + 6x 3 - 8x 4 + x 3)
10y 2 - 13y + 9
Subtract.
11.
4
3b 3c
12
SEE EXAMPLE
0.26r + 0.32r
2
14.
(16y 2 - 8y + 9) - (6y 2 - 2y + 7y)
(-7k 2 + 3) - (2k 2 + 5k - 1)
-9k 2 - 5k + 4
15. Geometry Write a polynomial that
represents the measure of angle ABD.
­nÊÊÊÊÊÊÓ>ÊÊx®
Ê>ÊÓÊ
Â
8a + 5a + 9
2
16–24
25–28
29–32
33–34
1
2
3
4
Quickly check key concepts.
Exercises: 24, 28, 30, 32, 33, 48
16. 4k 3 + 6k 2 + 9k 3
17. 5m + 12n 2 + 6n - 8m
18. 2.5a 4 - 8.1b 4 - 3.6b 4
19. 2d + 1 - d
20. 7xy - 4x y - 2xy
21. -6x 3 + 5x + 2x 3 + 4x 3
23. 3x 3 - 4 - x 3 - 1
24. 3b 3 - 2b - 1 - b 3 - b
5
5
22. x 2 + x + 3x + 2x 2
2
Extra Practice
Skills Practice p. S17
Application Practice p. S34
25.
27.
(2t 2 - 8t) + (8t 2 + 9t) 10t 2 + t
(x 5 - x) + (x 4 + x) x 5 + x 4
26.
28.
If you finished Examples 1–4
Basic 16–43, 45–51, 53–56,
63–72
Average 16–56, 58, 63–72
Homework Quick Check
PRACTICE AND PROBLEM SOLVING
Independent Practice
For
See
Exercises Example
If you finished Examples 1–2
Basic 16–28
Average 16–28, 58
­Ç>ÊÊ{®Â
Assign Guided Practice exercises
as necessary.
Auditory For Exercises
1–6, suggest to students
that they say each term out
loud as they do their homework to
hear the difference between like and
unlike terms.
-3x 2 - 11x + 3
(-7x 2 - 2x + 3) + (4x 2 - 9x)
(-2z 3 + z + 2z 3 + z) + (3z 3 - 5z 2)
3z 3 - 5z 2 + 2z
Subtract.
29.
31.
(t 3 + 8t 2) - (3t 3) -2t 3 + 8t 2
(5m + 3) - (6m 3 - 2m 2)
30.
-6m 3 + 2m 2 + 5m + 3
32.
(3x 2 - x) - (x 2 + 3x - x) 2x 2 - 3x
(3s 2 + 4s) - (-10s 2 + 6s) 13s 2 - 2s
33. Photography The measurements
of a photo and its frame are shown
in the diagram. Write a polynomial
that represents the width of the
photo. 4w 2 + 6w + 4
34. Geometry The length of a
rectangle is represented by
4a + 3b, and its width is
represented by 7a - 2b. Write
a polynomial for the perimeter
of the rectangle. 22a + 2b
ÈÜ ÓÊÊn
¶
Ü ÓÊÎÜÊÊÓ
7- 7 Adding and Subtracting Polynomials
507
24. 2b3 - 3b - 1
7. 23n3 + 3n + 15
8. 8q2 - 10.9q + 5.3
A1NL11S_c07_0504-0509.indd 507
9.
9x2
-x-6
10.
3x4
+ 8x3
16.
13k3
6/25/09 6:10:10 PM
+ 6k2
17. 12n2 + 6n - 3m
18. 2.5a4 - 11.7b4
19. d5 + 1
20. -4x2y + 5xy
21. 5x
22. 3x2 + 4x
23. 2x3 - 5
KEYWORD: MA7 Resources
Lesson 7-7
507
Exercise 53 involves
writing expressions
for the dimensions
of a rectangle. This exercise prepares
students for the Multi-Step Test Prep
on page 528.
41. -u + 3u +
3u + 6
43.
35. (2t - 7) + (-t + 2) t - 5
37. (4n - 2) - 2n 2n - 2
3
involves combining like terms.
No matter what order the terms
are combined in, the sum will
be the same. Yes; in polynomial
subtraction, the subtraction sign
is distributed among all terms in
the second polynomial, changing
all the signs to their opposites.
38. (-v - 7) - (-2v) v - 7
6x 2 - x - 1
(4x + 3x - 6) + (2x - 4x + 5)
(5u 2 + 3u + 7) - (u 3 + 2u 2 + 1)
2
39.
41.
2
(4m 2 + 3m) + (-2m 2) 2m 2 + 3m
4z 2 - 10z - 4
40.
(2z 2 - 3z - 3) + (2z 2 - 7z - 1)
42.
(-7h 2 - 4h + 7) - (7h 2 - 4h + 11)
-14h 2 - 4
43. Geometry The length of a rectangle is represented by 2x + 3, and its
width is represented by 3x + 7. The perimeter of the rectangle is 35 units.
Find the value of x.
_3 , or 1.5
2
44. Write About It If the parentheses are removed from (3m 2 - 5m) +
2
equivalent to the original? If the
44. Yes; the simplified (12m + 7m - 10), is the new expression
parentheses
are removed from (3m 2 - 5m) - (12m 2 + 7m - 10), is the new
form of both expressions
expression equivalent to the original? Explain.
is 15m 2 + 2m - 10.
No; the simplified
45. /////ERROR ANALYSIS///// Two students found the sum of the polynomials
form of the orig.
(-3n 4 + 6n 3 + 4n 2) and (8n 4 - 3n 2 + 9n). Which is incorrect? Explain
expression is
the error.
-9m 2 - 12m + 10,
and the simplified form
!
"
of the new expression is
,g -/g ,-g +)g
,g -/g ,-g +
-9m 2 + 2m - 10.
-)g ,,g +2g
1g -,g +2g
1
g
XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
45. B is incorrect. The student
.g -/g ,g +2g
.g -,g ,*,g +
incorrectly tried to combine 6n 3
and -3n 2, which are not like
terms, and 4n 2 and
Copy and complete the table by finding the missing polynomials.
9n, which are not
like terms.
x
53a.
2
36.
x
Polynomial 1
Polynomial 2
Sum
46.
x -6
3x - 10x + 2
4x - 10x - 4
47.
12x + 5
3x + 6
15x + 11
48.
x 4 - 3x 2 - 9
5x 4 + 8
6x 4 - 3x 2 - 1
49.
7x 3 - 6x - 3
6x + 14
7x 3 + 11
50.
2x + 5x
7x - 5x + 1
9x 3 + 1
x+x +6
3x 2 + 2x + 1
2
3
2
2
2x 2 + x - 5
51.
3
2
2
2
52. Critical Thinking Does the order in which you add polynomials affect the sum?
Does the order in which you subtract polynomials affect the difference? Explain.
53. This problem will prepare you for the Multi-Step Test Prep on page 528.
a. Ian plans to build a fenced dog pen. At first, he planned for the pen to be a
square of length x feet on each side, but then he decided that a square may not
be best. He added 4 feet to the length and subtracted 3 feet from the width.
Draw a diagram to show the dimensions of the new pen.
b. Write a polynomial that represents the amount of fencing that Ian will need for
the new dog pen. 4x + 2
c. How much fencing will Ian need if x = 15? 62 ft
7-7 PRACTICE A
7-7 PRACTICE C
Practice B
LESSON
7-7
7-7 PRACTICE B
508
Chapter 7 Exponents and Polynomials
12m 3 2m 2 3
1. 3m3 + 8m3 3 + m3 2m2
–7p 5 10pg + 5g
2. 2pg p5 12pg + 5g 6p5
3. 3k2 2k + 7
+
5x2 2x + 3y
4.
k2
3k2 k + 5
2
5. 11hz3 + 3hz2 + 8hz
+ 6x2 + 5x + 6y
+ 9hz3 + hz2 3hz
11x2 + 3x + 9y
20hz 3 + 4hz 2 + 5hz
2
6. (ab + 13b 4a) + (3ab + a + 7b)
2
3
7-7
3
5x 2x
2
7. (4x x + 4x) + (x x 4x)
12d2 + 3dx + x
2v5 3v4 8
9.
(4d2 + 2dx 8x)
(3v5 + 2v4 8)
16d 2 + dx + 9x
a
v5 5v4
11. (r2 + 8pr p) (12r2 2pr + 8p)
2
3
3
2
12. (un n + 2un ) (3un + n + 4un)
10.
Subtract (x2 + 8x − 4) − (3x2 − 3x + 2).
7 − 3(x + 8) + 4x
(x2 + 8x − 4) − (3x2 − 3x + 2)
2
33b 8
14. Darnell and Stephanie have competing refreshment stand businesses.
Darnell’s profit can be modeled with the polynomial c2 + 8c 100,
where c is the number of items sold. Stephanie’s profit can be modeled
with the polynomial 2c2 7c 200.
a. Write a polynomial that represents the difference between Stephanie’s
profit and Darnell’s profit.
c2 15c 100
b. Write a polynomial to show how much they can expect to earn if they
3c + c 300
3
= x − 17
= x2 − 3x2 + 8x + 3x − 4 − 2
= −2x2 + 11x − 6
1. What is being distributed in the linear expression on the left?
−3
2. What is being distributed in the polynomial subtraction on the right?
−1
The following are not like terms:
6/25/09 9:20:14 AM
−3x and 4x; 7 and −24
x 2 and −3x 2 ; 8x and 3x; −4 and −2
2x 3 + 2x + 7
9x 2 + 16x + 2
Rearrange terms so that like terms are together.
8x2 + 10x
Combine like terms.
Add (5y2 + 7y + 2) + (4y2 + y + 8).
(5y2 + 7y + 2 ) + (4y2 + y + 8 )
Identify like terms.
(5y2 + 4y2) + ( 7y + y ) + ( 2 + 8 )
Rearrange terms so that like terms are together.
9y2 + 8y + 10
Combine like terms.
no; same variable raised to different power
1. 4x and x4
3
3. 2z and 4x
yes; same variable raised to same power
3
no; different variable raised to same power
4. 2y2 + 3y + 7y + y2
7. (2x2 + 10x + 4) + (7x2 + 6x − 2)
Identify like terms.
3x2 + 5x2 + 4x + 6x
2. 5y and 7y
4. Identify the sets of like terms that were combined in the polynomial subtraction on the right.
5. 5x3 + 2x + 1 − 3x3 + 6
Add 3x2 + 4x + 5x2 + 6x.
3x2 + 4x + 5x2 + 6x
Determine whether the following are like terms. Explain.
3. Identify the sets of like terms that were combined in the expression on the left.
4x 4 + 2x − 2
Chapter 7
The following are like terms:
= x + 8x − 4 − 3x + 3x − 2
Complete the following based on the examples above.
9. (6x4 + 8x − 2) − (2x4 + 6x)
508
7-7 RETEACH
7-7
You can add or subtract polynomials by combining like terms.
2
Step 3: Combine all the sets of like terms.
11r 2 + 10pr 9p
13. Antoine is making a banner in the shape of a triangle. He
wants to line the banner with a decorative border. How long
will the border be?
2
= −3x + 4x + 7 − 24
5y4 + 8ay2 2y +
3un 2n un
2
Step 2: Rearrange so like terms are together.
y4 + 6ay2 y + a
Review for Mastery
LESSON
Step 1: Use the Distributive Property.
= 7 − 3x − 24 + 4x
(6y4 2ay 2 + y)
2
Connecting Concepts
Simplify 7 − 3(x + 8) + 4x.
Subtract.
8.
4ab + 20b 3a
A1NL11S_c07_0504-0509.indd
508
2
3
LESSON
The process for adding and subtracting polynomials is the same as the
process for simplifying linear expressions. Look at the connections below.
6. x − 3x5 + 2x4 − 5x5 − x
−8x 5 + 2x 4
8. (x3 − 6) + (9 − 2x2 + x3)
2x 3 − 2x 2 + 3
10. (3x2 − 9x) − (x + 2x3 − 4)
−2x 3 + 3x 2 − 10x + 4
2
3y + 10y
5. 8m4 + 3m 4m4
4
4m + 3m
7. (6x2 + 3x) + (2x2 + 6x)
8. (m2 10m + 5) + (8m + 2)
9. (6x3 + 5x) + (4x3 + x2 2x + 9)
10. (2y5 6y3 + 1) + (y5 + 8y4 2y3 1)
6. 12x5 + 10x4 + 8x4
12x5 + 18x4
8x2 + 9x
m2 2m + 7
10x3 + x2 + 3x + 9
3y5 + 8y4 8y 3
In Exercise 54, after
determining that the
missing term is a
y-term, encourage students to write
and solve an algebraic equation to
find the coefficient:
-12 + x - 6 = -15.
54. What is the missing term?
(-14y 2 + 9y 2 - 12y + 3) + (2y 2 +
-6y
- 6y - 2) = (-3y 2 - 15y + 1)
-3y
3y
6y
55. Which is NOT equivalent to -5t - t?
3
-(5t 3 + t)
(t 3 + 6t) - (6t 3 + 7t)
(2t 3 - 3t 2 + t) - (7t 3 - 3t 2 + 2t)
(2t 3 - 4t) - (-7t - 3t)
56. Extended Response Tammy plans to put a
wallpaper border around the perimeter of her
room. She will not put the border across the
doorway, which is 3 feet wide.
56b. 7; If x = 7,
a. Write a polynomial that represents the
Tammy will need
number of feet of wallpaper border that
Tammy will need. 6x + 3
6(7) + 3 = 45 feet
of wallpaper border.
b. A local store has 50 feet of the border that
However, if x = 8,
Tammy has chosen. What is the greatest
Tammy will need
whole-number value of x for which this
6(8) + 3 = 51 feet
amount would be enough for Tammy’s
of wallpaper border,
which is more than
c. Determine the dimensions of Tammy’s room for
the store has.
the value of x that you found in part b. 13 ft × 11 ft
63.
À
64.
­ÝÊÊ{®ÊvÌ
58. 2m 3 + 2m,
2m 3 + m
­ÓÝÊÊ£®ÊvÌ
Journal
Have students describe two different
examples in their descriptions.
57. Geometry The legs of the isosceles triangle at
right measure (x 3 + 5) units. The perimeter of the
triangle is (2x 3 + 3x 2 + 8) units. Write a polynomial
that represents the measure of the base of the
triangle. 3x 2 - 2
ÊÝ ÊÎ Êx
ÊÝ ÊÎ Êx
Have students write four different
polynomials, using any of the following as the variable part of the
terms: x, y, x2, y2. Instruct students to
pick any two of the polynomials and
find the sum. Then have them find
the difference of the remaining two
polynomials.
59. Write two polynomials whose difference is 4m 3 + 3m.
60. Write three polynomials whose sum is 4m 3 + 3m.
60. 2m 3 + m,
m 3 + m, m 3 + m
66–68. For graphs, see p. A27.
58. Write two polynomials whose sum is 4m 3 + 3m.
59. 5m 3 + 2m,
m3 - m
65.
CHALLENGE AND EXTEND
58–62.
given.
61. Write two monomials whose sum is 4m 3 + 3m. 4m 3, 3m
62. Write three trinomials whose sum is 4m 3 + 3m.
2m 3 + m 2 + m, m 3 + m 2 + m, m 3 - 2m 2 + m
SPIRAL REVIEW
Solve each inequality and graph the solutions. (Lesson 3-2)
63. d + 5 ≥ -2 d ≥ -7
64. 15 < m - 11 m > 26
65. -6 + t < -6 t < 0
7-7
Write each equation in slope-intercept form. Then graph the line described by
each equation. (Lesson 5-7)
1 x + 6 y = 1 x + 3 68. y = 4 (-x + 1)
66. 3x + y = 8 y = -3x + 8 67. 2y = _
2
4
_
1. 7m2 + 3m + 4m2 11m2 + 3m
y = -4x + 4
Simplify. (Lesson 7-3)
70. cd 4 · (c -5)
3
69. b 4 · b 7 b 11
d
_
4
c 14
71.
(-3z 6)2 9z 12
72.
(j 3k -5)3 · (k 2)4
7- 7 Adding and Subtracting Polynomials
LESSON
7-7
Problem Solving
7-7 PROBLEM SOLVING
1. There are two boxes in a storage unit.
The volume of the first box is 4x3 + 4x2
cubic units. The volume of the second
box is 6x3 − 18x2 cubic units. Write a
polynomial for the total volume of the
boxes.
A1NL11S_c07_0504-0509.indd two
509
10x 3 − 14x 2 cubic units
2. The recreation field at a middle school is
shaped like a rectangle with a length of
15x yards and a width of 10x − 3 yards.
Write a polynomial for the perimeter of
the field. Then calculate the perimeter if
x = 2.
50x − 6
94 yards
LESSON
7-7
Challenge
7-7 CHALLENGE
Polynomial Functions
A polynomial function is a function whose rule is a polynomial. For
example, P(x) = 2x3 4x2 + 3x 2 is a polynomial function.
Like other functions, polynomial functions can be evaluated by substituting
a value for the variable and simplifying.
4
1. For P(x) = 2x3 4x2 + 3x 2, find P(2).
A variable, or variable expression, can be substituted for the variable
in a function as well. For example, if P(x) = 5x + 3, then P(a) = 5a + 3.
Suppose P(x) = 3x 2. Then P(c) + P(c + 1) = 3c 2 + 3(c + 1) 2
3. Two cabins on opposite banks of a river
are 12x2 − 7x + 5 feet apart. One cabin
is 9x + 1 feet from the river. The other
cabin is 3x2 + 4 feet from the river.
Write the polynomial that represents
the width of the river where it passes
between the two cabins. Then calculate
the width if x = 3.
= 3c 2 + 3c + 3 2
= 6c 1
3. For P(x) = 10 4x, find P(d) + P(2d) P(3).
22 12d
9x 2 − 16x; 33 feet
4. For P(x) = 3x4 7x3 + 2x2 x + 8, find P(g2) + P(g).
The circle graph represents election results for the president of the
math team. Use the graph for questions 4–6. Select the best answer.
3g 8 7g 6 g 4 7g 3 + g 2 g + 16
Give an example of Q(x) and S(x) for each situation.
4. The angle value of Greg’s sector can
be modeled by x2 + 6x + 2. The
angle value of Dion’s sector can be
modeled by 7x + 20. Which polynomial
represents both sectors combined?
A x2 + x + 18
B
x2 + 13x + 22
5. Q(x) is a quartic binomial. S(x) is a quartic trinomial.
Q(x) + S(x) is a quartic trinomial.
Q(x) = x4 + 1; S(x) = x4 + x3 + 1
C 6x2 + 7x + 18
6. Q(x) is a cubic trinomial. S(x) is a cubic binomial.
Q(x) S(x) is a quadratic monomial.
D 7x2 + 6x + 22
Q(x) = x3 + x2 + 1; S(x) = x3 + 1
5. The sum of Greg and Lynn’s sectors
is 2x2 + 4x − 6. The sum of Max and
Dion’s sectors is 10x + 26. Which
polynomial represents how much greater
Greg and Lynn’s combined sectors are
than Max and Dion’s?
F 2x2 + 6x + 32
G 2x2 − 6x + 20
H
2x2 − 6x − 32
J 2x2 + 14x + 20
6. The sum of Lynn’s sector and Max’s
sector is 2x2 − 9x − 2. Max’s sector
can be modeled by 3x + 6. Which
polynomial represents the angle value of
Lynn’s sector?
2
2
A 2x − 6x + 4
C 2x − 12x + 8
B 2x2 − 6x − 4
D
2x2 − 12x − 8
7. Q(x) is a cubic trinomial. S(x) is a cubic binomial.
Q(x) + S(x) is a quadratic binomial.
k7
509
2. (r2 + s2) - (5r2 + 4s2)
-4r2 - 3s2
3. (10pq + 3p) +
(2pq - 5p + 6pq) 18pq - 2p
4. (14d 2 - 8) + (6d 2 - 2d + 1)
20d2 - 2d - 7
5. (2.5ab + 14b) (-1.5ab + 4b) 4ab + 10b
7/18/09 5:00:48 PM
4b 3 + b 6
2. For P(x) = 4x3 + x 6, find P(b).
j
_
9
6. A painter must add the areas
of two walls to determine the
amount of paint needed. The
area of the first wall is modeled by 4x2 + 12x + 9, and
the area of the second wall is
modeled by 36x2 - 12x + 1.
Write a polynomial that represents the total area of the two
walls. 40x2 + 10
Also available on transparency
Q(x) = x3 + x2 + 1; S(x) = x3 + x2
8. Q(x) is a quintic trinomial. S(x) is a quartic trinomial.
Q(x) + S(x) is a quintic polynomial with five terms.
Q(x) = x5 + x4 + 1; S(x) = x4 + x 3 + x2
Lesson 7-7
509
7-8
Organizer
Model Polynomial
Multiplication
Use with Lesson 7-8
Pacing:
1
day
2
1
__
Block 4 day
You can use algebra tiles to multiply polynomials. Use the length
and width of a rectangle to represent the factors. The area of the
rectangle represents the product.
Objective: Use algebra tiles to
Use with Lesson 7-8
model polynomial multiplication.
GI
Materials: algebra tiles
<
D@<I
KEY
REMEMBER
X
Online Edition
X
X
• The product of two values
with the same sign is
positive.
• The product of two values with
different signs is negative.
Countdown Week 16
Activity 1
Resources
Use algebra tiles to find 2(x + 1).
Algebra Lab Activities
7-8 Lab Recording Sheet
MODEL
Teach
ÝÊÊ£

Discuss

In Activity 3, remind students that
zero pairs are only those whose size
and shape are the same, but whose
colors are different.
Place the first factor in a column along the
left side of the grid. This will be the width
of the rectangle.
2(x + 1)
Place the second factor across the top of the
grid. This will be the length of the rectangle.
Ó
In Activity 1, show students that
they would get the same product if
they placed the first factor along the
top and the second factor along the
left side. This is because multiplication is commutative.
ALGEBRA

Fill in the grid with tiles that have the
same width as the tiles in the left column
and the same length as the tiles in the
top row.
The area of the rectangle inside the grid
represents the product.
Alternative Approach
x+x+1+1
2x + 2
Use the transparency mat and transparency algebra tiles (MK).
The rectangle has an area of 2x + 2, so 2(x + 1) = 2x + 2. Notice that this is the same product you
would get by using the Distributive Property to multiply 2(x + 1).
Try This
Use algebra tiles to find each product.
2. 2(2x + 1) 4x + 2
1. 3(x + 2) 3x + 6
510
510
Chapter 7
4. 3(2x + 2) 6x + 6
Chapter 7 Exponents and Polynomials
A1NL11S_c07_0510-0511.indd 510
KEYWORD: MA7 Resources
3. 3(x + 1) 3x + 3
6/25/09 9:38:46 AM
Close
Activity 2
Key Concept
Use algebra tiles to find 2x(x - 3).
MODEL
ÝÊÊÎ

ÓÝ

When multiplication is being modeled with algebra tiles, the dimensions of a rectangle represent the
factors, and the area of the rectangle
represents the product. The area of
the rectangle can sometimes be simplified by removing zero pairs.
ALGEBRA
Place tiles to form the length and width
of a rectangle and fill in the rectangle.
The product of two values with the same
sign (same color) is positive (yellow).
The product of two values with different
signs (different colors) is negative (red).
2x(x - 3)
The area of the rectangle inside the grid
represents the product.
x2 + x2 - x - x - x - x - x - x
Assessment
The rectangle has an area of 2x 2 - 6x, so
2x(x - 3) = 2x 2 - 6x.
Journal When using algebra tiles
to model polynomial multiplication,
have students explain how to determine which tiles should be placed
inside the multiplication grid to form
the rectangle (product). Explanations
should include how to determine
sizes, shapes, and colors of tiles.
2x 2 - 6x
Try This
Use algebra tiles to find each product.
5. 3x(x - 2)
6. x(2x - 1)
3x 2 - 6x
2x 2 - x
7. x(x + 1)
x2+x
8. (8x + 5)(-2x)
-16x 2 - 10x
Activity 3
Use algebra tiles to find (x + 1)(x - 2).
MODEL
ÝÊÊÓ

ÝÊÊ£

ALGEBRA
Place tiles for each factor to form
the length and width of a rectangle.
Fill in the grid and remove any zero
pairs.
The area inside the grid represents
the product.
The remaining area is x 2 - x - 2, so
(x + 1)(x - 2) = x 2 - x - 2.
(x + 1)(x - 2)
x2 - x - x + x - 1 - 1
x2 - x - 1 - 1
x2 - x - 2
Try This
Use algebra tiles to find each product.
x2-x-6
9. (x + 2)(x - 3)
x 2 + 2x - 3
10. (x - 1)(x + 3)
x 2 - 5x + 6
11. (x - 2)(x - 3)
x 2 + 3x + 2
12. (x + 1)(x + 2)
7- 8 Algebra Lab
A1NL11S_c07_0510-0511.indd 511
511
6/25/09 9:39:12 AM
7-8 Algebra Lab
511
7-8
Organizer
Multiplying
Polynomials
7-8
1
Block __
day
2
GI
Objective: Multiply polynomials.
<
D@<I
Why learn this?
You can multiply polynomials
to write expressions for areas,
such as the area of a dulcimer.
(See Example 5.)
Objective
Multiply polynomials.
Online Edition
Tutorial Videos, Interactivity
Countdown Week 17
To multiply monomials and
polynomials, you will use some of the
properties of exponents that you learned earlier in this chapter.
EXAMPLE
Warm Up
1
A (5x 2)(4x 3)
Simplify.
1.
32
9
3. 102
Multiplying Monomials
Multiply.
2.
24
16
100
4. 23 · 24
6. (53)2
5. y5 · y4
27
7. (x2)4
56
8. -4(x - 7)
y9
B
x8
-4x + 28
Also available on transparency
When multiplying
powers with the
same base, keep the
exponents.
C
x2 · x3 = x2+3 = x5
(5x 2)(4x 3)
(5 · 4)(x 2 · x 3)
Group factors with like bases together.
20x 5
Multiply.
(-3x 3y 2)(4xy 5)
(-3x 3y 2)(4xy 5)
(-3 · 4)(x 3 · x)(y 2 · y 5)
Group factors with like bases together.
-12x 4y 7
Multiply.
(_12 a b)(a c )(6b )
(_12 a b)(a c )(6b )
(_12 · 6)(a · a )(b · b )(c )
3
2 2
3
2 2
3
2
2
2
2
2
Group factors with like bases together.
3a 5b 3c 2
Student A: What is u times r
times r?
Multiply.
Multiply.
1a. (3x 3)(6x 2)
Student B: ur2.
18x
Student A: No, I’m not!
5
1b. (2r 2t)(5t 3)
2 4
10r t
(
)
1 x 2y (12x 3z 2) y 4z 5
1c. _
( )
3
5 5 7
4x y z
To multiply a polynomial by a monomial, use the Distributive Property.
EXAMPLE
2
Multiplying a Polynomial by a Monomial
Multiply.
A 5(2x 2 + x + 4)
5 (2x 2 + x + 4)
(5)2x 2 + (5)x + (5)4
10x 2 + 5x + 20
512
Distribute 5.
Multiply.
Chapter 7 Exponents and Polynomials
1 Introduce
A1NL11S_c07_0512-0519.indd
e x p l o r512
at i o n
7-8
Motivate
Multiplying Polynomials
You will need a graphing calculator for this Exploration. As you
work through the Exploration, try to ﬁnd a rule for multiplying a
monomial and a polynomial.
1. You can use your calculator to explore
the relationship between x 2x 2 ⫹ 3 and 2x 3 ⫹ 3x. Press 9 and enter
2
3
x 2x ⫹ 3 as Y1. Then enter 2x ⫹ 3x
as Y2.
/
2. Press ND '2!0( to view a table of
values for the two expressions.
Use the arrow keys to scroll up
and down the table. What do you
notice about the values of Y1 and Y2 for
each value of x ? Use this information to
2
3
make a conjecture about x 2x ⫹ 3 and 2x ⫹ 3x.
Use a calculator to predict whether each equation is true.
3. 2x 5x ⫹ 9 ⫽ 10x 2 ⫹ 18x
KEYWORD: MA7 Resources
4. 3x 3 2x ⫹ 9 ⫽ 6x 4 ⫹ 9
Chapter 7
Have students explain how to find the area of a
rectangle with length x and width 10.
A = w, so A = (x)(10) = 10x
Explain to students that in this lesson they will
learn to describe area when the dimensions are
polynomials.
5. 4x 2 3x ⫺ 5 ⫽ 12x 3 ⫺ 20x 2 6. 10x 3 3x 2 ⫺ 2x ⫽ 30x 5 ⫺ 20x 4
THINK AND DISCUSS
7. Describe how to multiply a monomial and a polynomial
512
Ask students to explain how to find the area of a
rectangle with length 5 cm and width 7 cm.
A = w, so A = (5)(7) = 35 cm2
Explorations and answers are provided in
Alternate Openers: Explorations Transparencies.
6/25/09 9:40:02 AM
Multiply.
""
B 2x y (3x - y)
(2x 2y)(3x - y)
(2x 2y)3x + (2x 2y)(-y)
When a binomial is raised to a
power, many students make the
following mistake:
(x + 4)2 = x2 + 42. Encourage
students to write the expression as
the product of two binomials first
and then multiply. For example,
(x + 4)2 = (x + 4)(x + 4) =
x2 + 4x + 4x + 16 = x2 + 8x + 16.
Distribute 2x 2y.
(2 · 3)(x 2 · x)y + 2 (-1)(x 2)(y · y)
3
Group like bases together.
6x y - 2x y
2 2
Multiply.
C 4a (a 2b + 2b 2)
4a (a 2b + 2b 2)
(4a) a 2b + (4a)2b 2
Distribute 4a.
(4)(a · a 2)(b) + (4 · 2)(a)(b 2)
Group like bases together.
4a b + 8ab
Multiply.
3
Ê,,",
,/
2
2
Example 1
Multiply.
2a. 2 (4x 2 + x + 3)
2b. 3ab (5a 2 + b)
15a 3b + 3ab 2
8x 2 + 2x + 6
2c. 5r 2s 2 (r - 3s)
5r 3s 2 - 15r 2s 3
To multiply a binomial by a binomial, you can apply the Distributive Property
more than once:
(x + 3)(x + 2) = x (x + 2) + 3(x + 2)
Distribute.
Multiply.
A. (6y3)(3y5)
18y8
B. (3mn2)(9m2n)
(
27m3n3
)
1
C. _ s2t2 (st)(-12st2)
4
-3s4t5
Example 2
= x (x + 2) + 3(x + 2)
Multiply.
= x (x) + x (2) + 3 (x) + 3 (2)
Distribute again.
= x 2 + 2x + 3x + 6
Multiply.
= x 2 + 5x + 6
Combine like terms.
A. 4(3x2 + 4x - 8)
12x2 + 16x - 32
B. 6pq(2p - q) 12p2q - 6pq2
1
C. _ x2y(6xy + 8x2y2)
2
3x3y2 + 4x4y3
Another method for multiplying binomials is called the FOIL method.
F
1. Multiply the First terms.
O
2. Multiply the Outer terms.
INTERVENTION
I
Questioning Strategies
3. Multiply the Inner terms.
EX AM P LE
L
4. Multiply the Last terms.
F
• When multiplying monomials with
exponents, why do you add the
exponents?
O
I
EX AM P LE
L
7- 8 Multiplying Polynomials
513
2
• Will a monomial times a binomial
always be a binomial?
• Will a monomial times a trinomial
always be a trinomial? Explain.
2 Teach
A1NL11S_c07_0512-0519.indd 513
Inclusion When using the
FOIL method, encourage
students to write the product after drawing each arrow rather
than drawing all of the arrows at
once. State: “Draw the F arrow and
write the product, draw the O arrow
and write the product,” and so on.
7/18/09 4:59:16 PM
Guided Instruction
Review the multiplication properties of
exponents before beginning this lesson.
Encourage students to write each step
carefully to avoid making mistakes when
distributing. Arrows can be drawn to help
keep track of the terms that have been
multiplied. Remind students that all of the
methods presented in this lesson will give
the method they are most comfortable
with.
1
• How is multiplying monomials
Through Visual Cues
Show students the “FOIL face” to help
them keep track of which terms to
multiply.
­ÝÊÊÎ®­ÝÊÊÓ®
Lesson 7-8
513
EXAMPLE
3
Multiplying Binomials
Multiply.
A (x + 2)(x - 5)
(x + 2 ) (x - 5 )
Example 3
Multiply.
A. (s + 4)(s - 2)
B. (x - 4)2
s2 + 2s - 8
Distribute.
x (x) + x (-5) + 2(x) + 2(-5)
Distribute again.
x - 5x + 2x - 10
Multiply.
x 2 - 3x - 10
Combine like terms.
2
x2 - 8x + 16
C. (8m2 - n)(m2 - 3n)
In the expression
(x + 5)2, the base
is (x + 5).
8m4 - 25m2n + 3n2
B (x + 5)
2
(x + 5)2 =
(x + 5)(x + 5)
(x + 5)(x + 5)
(x · x) + (x · 5) + (5 · x) + (5 · 5)
Write as a product of two binomials.
x + 5x + 5x + 25
Multiply.
x 2 + 10x + 25
Combine like terms.
Use the FOIL method.
2
INTERVENTION
C (3a 2 - b)(a 2 - 2b)
Questioning Strategies
EX A M P L E
x (x - 5) + 2(x - 5)
3a 2(a 2) + 3a 2(-2b) - b (a 2) - b (-2b) Use the FOIL method.
3
3a 4 - 6a 2b - a 2b + 2b 2
Multiply.
3a - 7a b + 2b
Combine like terms.
4
• What terms from FOIL can often be
combined?
2
2
3a. (a + 3)(a - 4)
• What do the signs from the last
terms in each binomial tell you
a - a - 12
2
3b. (x - 3)2
x - 6x + 9
2
3c. (2a - b 2)(a + 4b 2)
2a 2 + 7ab 2 - 4b 4
To multiply polynomials with more than two terms, you can use the Distributive
Property several times. Multiply (5x + 3) by (2x 2 + 10x - 6):
Math Background In
Example 3C, remind
students that the terms
6a2b and -ba2 are like terms due
to the Commutative Property of
Multiplication.
(5x + 3)(2x 2 + 10x - 6) = 5x (2x 2 + 10x - 6) + 3 (2x 2 + 10x - 6)
= 5x (2x 2 + 10x - 6) + 3(2x 2 + 10x - 6)
= 5x (2x 2) + 5x (10x) + 5x (-6) + 3(2x 2) + 3(10x) + 3(-6)
= 10x 3 + 50x 2 - 30x + 6x 2 + 30x - 18
= 10x 3 + 56x 2 - 18
You can also use a rectangle model to multiply polynomials with more than
two terms. This is similar to finding the area of a rectangle with length
(2x 2 + 10x - 6) and width (5x + 3):
5x
2x 2
+ 10x
-6
10x 3
50x 2
-30x
30x
-18
+3
6x
2
Write the product of the monomials
in each row and column.
To find the product, add all of the terms inside the rectangle by combining
like terms and simplifying if necessary.
10x 3 + 6x 2 + 50x 2 + 30x - 30x - 18
10x 3 + 56x 2 - 18
514
Chapter 7 Exponents and Polynomials
Through Modeling
A rectangle model can also be used to display the FOIL method.
A1NL11S_c07_0512-0519.indd 514
x
+3
x
x2
3x
+2
2x
6
This model shows that (x + 2)(x + 3) =
x2 + 3x + 2x + 6 = x2 + 5x + 6.
This rectangle model is useful because it
can also be used in the next chapter for
factoring polynomials.
514
Chapter 7
6/25/09 9:40:42 AM
Another method that can be used to multiply polynomials with more than
two terms is the vertical method. This is similar to methods used to
multiply whole numbers.
2x 2 + 10x - 6
Example 4
×
5x + 3
−−−−−−−−−−−
−−−−
6x 2 + 30x - 18
3
2
+
10x + 50x - 30x
−−−−−−−−−−−−−−−−−
10x 3 + 56x 2 + 0x - 18
10x 3 + 56x 2
- 18
EXAMPLE
4
Multiply.
Multiply each term in the top polynomial by 3.
Multiply each term in the top polynomial by 5x,
and align like terms.
Combine like terms by adding vertically.
Simplify.
A. (x - 5)(x2 + 4x - 6)
x3 - x2 - 26 x + 30
B. (2x - 5)(-4x2 - 10x + 3)
-8x3 + 56 x - 15
C. (x + 3)3 x3 + 9x2 + 27x + 27
D. (3x + 1)(x3 + 4x2 - 7) 3x4 +
13x3 + 4x2 - 21x - 7
Multiplying Polynomials
Multiply.
A (x + 2)(x 2 - 5x + 4)
A polynomial with
m terms multiplied
by a polynomial
with n terms has a
product that, before
simplifying, has mn
terms. In Example
4A, there are 2 · 3,
or 6, terms before
simplifying.
(x + 2)(x 2 - 5x + 4)
x (x 2 - 5x + 4) + 2(x 2 - 5x + 4)
Distribute.
x (x
Distribute again.
) + x (-5x) + x (4) + 2(x ) + 2(-5x) + 2(4)
2
2
x 3 + 2x 2 - 5x 2 - 10x + 4x + 8
Simplify.
x 3 - 3x 2 - 6x + 8
Combine like terms.
INTERVENTION
Questioning Strategies
EX AM P LE
B (3x - 4)(-2x 3 + 5x - 6)
(3x - 4)(-2x 3 + 5x - 6)
-2x 3 + 0x 2 + 5x - 6
×
3x - 4
−−−−−−−−−−−−−−−−
−−−−−
3
2
8x + 0x - 20x + 24
+ -6x 4 + 0x 3 + 15x 2 - 18x
−−−−−−−−−−−−−−−−−−−−−
-6x 4 + 8x 3 + 15x 2 - 38x + 24
4
• How is multiplying a polynomial by
a binomial similar to multiplying a
binomial by a binomial? How is it
different?
Add 0x 2 as a placeholder.
Multiply each term in the top
polynomial by -4.
Multiply each term in the top polynomial
by 3x, and align like terms.
Combine like terms by adding vertically.
students to find similarities
between polynomials and
real numbers. Discuss the types of
operations you can use with each.
C (x - 2)3
⎡⎣(x - 2)(x - 2)⎤⎦(x - 2)
Write as the product of three
binomials.
⎡⎣x · x + x (-2) + (-2) x + (-2)(-2)⎤⎦(x - 2)
Use the FOIL method on the
first two factors.
(x 2 - 2x - 2x + 4)(x - 2)
(x 2 - 4x + 4)(x - 2)
(x - 2)(x 2 - 4x + 4)
x (x 2 - 4x + 4) + (-2)(x 2 - 4x + 4)
x (x 2) + x (-4x) + x (4) + (-2)(x 2) +
(-2)(-4x) + (-2)(4)
Multiply.
Combine like terms.
Use the Commutative
Property of Multiplication.
Distribute.
Distribute again.
x 3 - 4x 2 + 4x - 2x 2 + 8x - 8
Simplify.
x 3 - 6x 2 + 12x - 8
Combine like terms.
7- 8 Multiplying Polynomials
A1NL11S_c07_0512-0519.indd 515
515
7/20/09 5:23:11 PM
Lesson 7-8
515
Multiply.
D (2x + 3)(x 2 - 6x + 5)
Example 5
The width of a rectangular
prism is 3 feet less than the
height, and the length of the
prism is 4 feet more than the
height.
x2
- 6x
+5
2x
2x 3
-12x 2
10 x
+3
3x 2
-18x
15
Write the product of the monomials
in each row and column.
2x 3 + 3x 2 - 12x 2 - 18x + 10x + 15
2x 3 - 9x 2 - 8x + 15
Add all terms inside the rectangle.
Combine like terms.
Multiply.
x 3 - x 2 - 6x + 18
4a. (x + 3)(x 2 - 4x + 6)
a. Write a polynomial that
represents the area of the
base of the prism.
h2 + h - 12
4b. (3x + 2)(x 2 - 2x + 5)
LÓÊÊÊÊ£
3x 3 - 4x 2 + 11x + 10
b. Find the area of the base
when the height is 5 feet.
18 ft2
EXAMPLE
5
Music Application

A dulcimer is a musical instrument that
is sometimes shaped like a trapezoid.
L£ÊÊÓÊÊ£
A Write a polynomial that represents the area of the dulcimer shown.
1h b +b
A=_
( 1 2)
2
1 h ⎡(2h - 1) + (h + 1)⎤
=_
⎣
⎦
2
1 h (3h)
=_
2
3
_
= h2
2
The area is represented by __32 h 2.
INTERVENTION
Questioning Strategies
EX A M P L E
5
• What types of polynomials did you
multiply?
Write the formula for area of a trapezoid.
Substitute 2h - 1 for b 1 and h + 1 for b 2.
Combine like terms.
Simplify.
B Find the area of the dulcimer when the height is 22 inches.
3 h2
A=_
2
_
= 3 (22)2
2
3
_
= (484) = 726
2
The area is 726 square inches.
5a. x 2 - 4x
Use the polynomial from part a.
Substitute 22 for h.
5. The length of a rectangle is 4 meters shorter than its width.
a. Write a polynomial that represents the area of the rectangle.
b. Find the area of the rectangle when the width is 6 meters.
5b. 12 m2
THINK AND DISCUSS
1. Compare the vertical method for
multiplying polynomials with the vertical
method for multiplying whole numbers.
2. GET ORGANIZED Copy and complete the
graphic organizer. In each box, multiply two
polynomials using the given method.
516
(y + 2)(y - 5) y2 - 3y - 10
(x + 5y)(xy + 4x + 7y)
x2y + 4x2 + 27xy + 5xy2 + 35y2
516
Chapter 7
ÕÌ«Þ}Ê*Þ>Ã
,iVÌ>}i
`i
6iÀÌV>
iÌ`
A1NL11S_c07_0512-0519.indd 516
Ask students to multiply the following
using any of the methods learned in this
lesson.
(5r2s)(9rs) 45r3s2
3x2(x3 - 10) 3x5 - 30x2
"
iÌ`
Chapter 7 Exponents and Polynomials
3 Close
Summarize
ÃÌÀLÕÌÛi
*À«iÀÌÞ
and INTERVENTION
Diagnose Before the Lesson
7-8 Warm Up, TE p. 512
Monitor During the Lesson
Check It Out! Exercises, SE pp. 512–516
Questioning Strategies, TE pp. 513–516
Assess After the Lesson
7-8 Lesson Quiz, TE p. 519
Alternative Assessment, TE p. 519
1. Possible answer: Both numbers and
polynomials are set up in 2 rows and
require you to multiply each item in the
top row by an item in the bottom row.
In the end, you add vertically to get
the answer. When you are multiplying
polynomials, the items are monomial
terms. When you are multiplying numbers, the items are digits.
2. See p. A7.
6/25/09 9:41:17 AM
7-8
Exercises
7-8 Exercises
KEYWORD: MA11 7-8
KEYWORD: MA7 Parent
GUIDED PRACTICE
Assignment Guide
Multiply.
SEE EXAMPLE
1
SEE EXAMPLE
(-5mn 3)(4m 2n 2)
1 r 4t 3
3. (6rs 2)(s 3t 2) _
2
4.
(_13 a )(12a)
5.
(-3x 4y 2)(-7x 3y)
6. (-2pq 3)(5p 2q 2)(-3q 4)
3
8. 3ab (2a 2 + 3b 3)
9. 2a 3b(3a 2b + ab 2)
10. -3x (x - 4x + 6)
11. 5x y (2xy - y)
12. 5m 2n 3 · mn 2(4m - n)
13. (x + 1)(x - 2)
14. (x + 1)2
15. (x - 2)2
16.
SEE EXAMPLE 4
2
(y - 3)(y - 5)
17.
19. (x + 5)(x 2 - 2x + 3)
3
18. (m 2 - 2mn)(3mn + n 2)
(4a 3 - 2b)(a - 3b 2)
20. (3x + 4)(x 2 - 5x + 2)
21. (2x - 4)(-3x 3 + 2x - 5)
22. (-4x + 6)(2x 3 - x 2 + 1) 23. (x - 5)(x 2 + x + 1)
p. 515
SEE EXAMPLE
5
2
p. 514
5
p. 516
Assign Guided Practice exercises
as necessary.
2.
7. 4(x 2 + 2x + 1)
2
p. 512
)
(2x 2)(7x 4)
p. 512
SEE EXAMPLE
(
1.
24. (a + b)(a - b)(b - a)
25. Photography The length of a rectangular photograph is 3
inches less than twice the width. 2x 2 - 3x
a. Write a polynomial that represents the area of the photograph.
Homework Quick Check
Quickly check key concepts.
Exercises: 30, 42, 46, 52, 56, 62,
69
Ý
in2
PRACTICE AND PROBLEM SOLVING
Multiply.
Independent Practice
For
See
Exercises Example
26–34
35–43
44–52
53–61
62
26.
1
2
3
4
5
29.
32.
Extra Practice
Skills Practice p. S17
Application Practice p. S34
(3x 2)(8x 5)
(-2a 3)(-5a)
(7x 2)(xy 5)(2x 3y 2)
27.
(-2r 3s 4)(6r 2s)
30.
(6x 3y 2)(-2x 2y)
(
2
3
2
)
1 x 2z 3 y 3z 4
28. (15xy 2) _
( )
3
31. (-3a 2b)(-2b 3)(-a 3b 2)
)(a b c)(3ab
2
9(2x - 5x)
5s 2t 3(2s - 3t 2)
-2a 2b 3(3ab 2 - a 2b)
33. (-4a bc
3
)
4 5
c
34. (12mn
)(2m n)(mn)
2
3x(9x - 4x)
2
2
35. 9s(s + 6)
36.
38. 3(2x 2 + 5x + 4)
39.
41. -5x (2x 2 - 3x - 1)
42.
44. (x + 5)(x - 3)
45. (x + 4)2
46. (m - 5)2
47. (5x - 2)(x + 3)
48. (3x - 4)2
49. (5x + 2)(2x - 1)
50. (x - 1)(x - 2)
51. (x - 8)(7x + 4)
52. (2x + 7)(3x + 7)
53.
56.
59.
(x + 2)(x 2 - 3x + 5)
(x - 3)(x 2 - 5x + 6)
(x - 2)(x 2 + 2x + 1)
54.
57.
60.
37.
40. x 2y 3 · 5x 2y (6x + y 2)
1. 14x6
2. -20m3n5
3. 3r5s5t5
4. 4a6
43. -7x 3y · x 2y 2(2x - y)
(2x + 5)(x 2 - 4x + 3)
(2x 2 - 3)(4x 3 - x 2 + 7)
(2x + 10)(4 - x + 6x 3)
If you finished Examples 1–5
Basic 26–69, 75–84, 87–89,
98–104
Average 26–65, 70–80 even,
82–93, 98–104
82–104
ÓÝÊÊÎ
b. Find the area of the photograph when the width is 4 inches.
20
If you finished Examples 1–3
Basic 26–52, 70–78
Average 26–52, 70–78
5. 21x7y3
6. 30p3q9
7. 4x2 + 8x + 4
55. (5x - 1)(-2x 3 + 4x - 3)
8. 6a3b + 9ab4
9. 6a5b2 + 2a4b3
58. (x - 4)3
10. -3x3 + 12x2 - 18x
61. (1 - x)
11. 10x3y4 - 5x2y2
3
12. 20m4n5 - 5m3n6
62. Geometry The length of the rectangle at right is 3 feet longer
than its width.
x 2 + 3x
a. Write a polynomial that represents the area of the rectangle.
b. Find the area of the rectangle when the width is 5 feet. 40 ft2
13. x2 - x - 2
ÝÊÊÎ
14. x2 + 2x + 1
Ý
63. A square tabletop has side lengths of (4x - 6) units. Write a polynomial that
represents the area of the tabletop. 16x 2 - 48x + 36
7- 8 Multiplying Polynomials
15. x2 - 4x + 4
16. y2 - 8y + 15
A1NL11S_c07_0512-0519.indd
17.5174a4
- 2ab -
12a3b2
+
6b3
18. 3m3n - 5m2n2 - 2mn3
19. x3 + 3x2 - 7x + 15
20. 3x3 - 11x2 - 14x + 8
21.
+
18x + 20
-6x4
12x3
+
4x2
-
22. -8x4 + 16x3 - 6x2 4x + 6
23. x3 - 4x2 - 4x - 5
24. -a3 + a2b + ab2 - b3
26. 24x7
38. 6x2 + 15x + 12
27. -12r5s5
39. 10s3t3 - 15s2t5
28.
5x3y5z7
29.
10a4
40. 30x5y4 + 5x4y6
41.
-10x3
+
15x2
42. -6a3b5 + 2a4b4
31. -6a5b6
43. -14x6y3 + 7x5y4
44. x2 + 2x - 15
33. -12a7b7c8
45. x2 + 8x + 16
32.
34.
24m4n4
35. 9s2 + 54s
36.
18x2
- 45x
37. 27x3 - 12x2
6/25/09 9:41:45 AM
+ 5x
30. -12x5y3
14x6y7
517
46. m2 - 10m + 25
47. 5x2 + 13x - 6
48. 9x2 - 24x + 16
49–61. See p. A27.
KEYWORD: MA7 Resources
Lesson 7-8
517
Exercise 64 involves
multiplying polynomials. This exercise
prepares students for the Multi-Step
Test Prep on page 528.
64. This problem will prepare you for the Multi-Step Test Prep on page 528.
a. Marie is creating a garden. She designs a rectangular garden with a length of
(x + 4) feet and a width of (x + 1) feet. Draw a diagram of Marie’s garden with
the length and width labeled.
b. Write a polynomial that represents the area of Marie’s garden. x 2 + 5x + 4
c. What is the area when x = 4? 40 ft2
64a.
65. Copy and complete the table below.
x
A
Degree
of A
B
Degree
of B
A·B
2x 2
2
3x 5
5
6x 7
3
2x 2 + 1
2
2
x2 - x
2
4
1
x 3 - 2x 2 + 1
3
4
x
65b. x4 - x3 + 2x2 - 2x
c. x4 - 5x3 + 6x2 + x - 3
70. 6a9
71. 2x2 - 7x - 30
a.
5x 3
b.
x2 + 2
c.
x-3
72. 3g2 + 14g - 5
73.
8x2
74.
x2
- 16xy +
6y2
75. 6x2 - 9x - 6
66.
77. x3 + 3x2
67.
79. 2x3 - 7x2 - 10x + 24
-
-
ab2
+
12x 2 + 12x + 3
{Ý
ÝÊÊx
8x + 12x
Sports
b3
ÝÊÊx
ÓÝÊÊ£
2
x 2 - 10x + 25
69. Sports The length of a regulation team handball court is twice
its width.
a. Write a polynomial that represents the area of the court. 2x 2
81. 8p3 - 36p2q + 54pq2 - 27q3
82a.
5
68.
Î­ÓÝÊÊ£®
ÓÝÊÊÎ
78. x3 + 3x2 + 2x
80.
10x + 5x
3
Geometry Write a polynomial that represents the area of each rectangle.
76. x2 - 6x - 40
a2b
7
5
d. Use the results from the table to complete the following: The product of a
polynomial of degree m and a polynomial of degree n has a degree of . m + n
-9
a3
Degree
of A · B
x
Ý
ÓÝ
b. The width of a team handball court is 20 meters. Find the
area of the court. 800 m2
x
83. Possible answer: Each letter in
FOIL represents a pair of terms
in a certain position within the
factors. The letters must account
for every pairing of terms while
describing first, outside, inside,
and last positions. This is only
possible with 2 binomials.
Multiply.
Team handball is a
game with elements of
originated in Europe in
the 1900s and was first
played at the Olympics
in 1936 with teams
of 11 players. Today, a
handball team consists of
seven players—six court
players and one goalie.
7-8 PRACTICE C
__________________
(1.5a 3)(4a 6)
71. (2x + 5)(x - 6)
72.
73.
(4x - 2y)(2x - 3y)
74. (x + 3)(x - 3)
75. (1.5x - 3)(4x + 2)
76. (x - 10)(x + 4)
77. x (x + 3)
78. (x + 1)(x 2 + 2x)
79. (x - 4)(2x 2 + x - 6)
80.
2
(a + b)(a - b)2
81.
(3g - 1)(g + 5)
(2p - 3q)3
82. Multi-Step A rectangular swimming pool is 25 feet long and 10 feet wide. It is
surrounded by a fence that is x feet from each side of the pool.
a. Draw a diagram of this situation.
b. Write expressions for the length and width of the fenced region. (Hint: How
much longer is one side of the fenced region than the corresponding side of
the pool?) 2x + 25; 2x + 10
c. Write an expression for the area of the fenced region. 4x 2 + 70x + 250
83. Write About It Explain why the FOIL method can be used to multiply only two
binomials at a time.
7-8 PRACTICE A
_______________________________________
70.
__________________
Practice B
7-8 PRACTICE B
Multiplying Polynomials
LESSON
7-8
518
Chapter 7 Exponents and Polynomials
Multiply.
1. (6m 4 ) (8m 2 )
2. (5x 3 ) (4xy 2 )
3. (10s 5 t)(7st 4 )
48m 6
20x4 y2
70s6 t 5
2
4. 4(x + 5x + 6)
5. 2x(3x 4)
4x2 + 20x + 24
7. (x + 3) (x + 4)
9. (x 2) (x 5)
x2 12x + 36
2x2 + 17x + 30
x2 7x + 10
13. (x + 4) (x 2 + 3x + 5)
x3 + 7x2 + 17x + 20
12. (a 2 + b 2 ) (a + b)
5m 4 + m 3 n + 15m
+ 3n
14. (3m + 4) (m 2 3m + 5)
2
Multiply (5x − 4) (3x + x − 8).
A1NL11S_c07_0512-0519.indd 518
a3 + a 2 b + ab 2 + b 3
15. (2x 5)(4x 2 3x + 1)
3m 3 5m2 + 3m
+ 20
There are several methods that can be used to multiply polynomials,
depending on the number of terms. There is one procedure that can always
be used, no matter how many terms there are. It is shown in the example
below.
21x3 y + 28xy2
+ 14xy
11. (m 3 + 3) (5m + n)
10. (2x + 5) (x + 6)
7-8
6. 7xy(3x + 4y + 2)
6x2 8x
8. (x 6) (x 6)
x2 + 7x + 12
Name _______________________________________ Date __________________ Class__________________
LESSON
2
8x3 26x2 + 17x 5
(5x − 4) (3x2 + x − 8)
1
Use the Distributive Property.
2
Collect like terms.
5x(3x2 + x − 8) −4(3x2 + x − 8)
15x3 + 5x2 − 40x − 12x2 − 4x + 32
15x3 + 5x2 − 40x − 12x2 − 4x + 32
15x3 + (5x2 − 12x2) + (−40x − 4x) + 32
of the rectangle.
b. Find the area of the rectangle when the
width is 4 inches.
2
w + 3w
28 in2
a. Write a polynomial that represents the area
of the rectangle.
width is 10 centimeters.
15x3 − 7x2 − 44x + 32
3
Simplify by combining like terms.
Use the procedure shown above to answer each of the following.
17. The length of a rectangle is 8 centimeters less than 3 times the width.
b. Find the area of the rectangle when the
15x3 + (5x2 − 12x2) + (−40x − 4x) + 32
2
3w 8w
220 cm2
18. Write a polynomial to represent the volume of the rectangular prism.
1. Multiplication was used six times in step 1. How many times would it be used if two
binomials were being multiplied?
4
2. In step 2, how do you know that 5x2 and −12x2 are like terms?
They have the same exponent on the same variable.
3. In step 3, how do you know the expression is completely simplified?
There are no like terms.
1 x3 5 x2 13x + 60
2
2
−6x 5 + 12x 4 − 3x 3
6. (7x + 2) (x − 3)
7x 2 − 19x − 6
518
Chapter 7
Multiplying Polynomials
7-8 RETEACH
Multiply (3a 2 b) (4ab3 ).
(3a 2 b) (4ab 3 )
(3 • 4) (a 2 • a) (b • b 3 )
Rearrange so that the constants and the variables with the same
bases are together.
12a 3 b 4
Multiply.
To multiply a polynomial by a monomial, distribute the monomial to each term in the
polynomial.
2x(x 2 + 3x + 7)
(2x)x 2 + (2x)3x + (2x)7
Distribute.
2x 3 + 6x 2 + 14x
Multiply.
Multiply.
1. (5x 2 y 3 ) (2xy)
2. (2xyz) (4x 2 yz)
3. (3x) (x 2 y 3 )
10x3 y4
8x3 y2 z 2
3x3 y3
Fill in the blanks below. Then finish multiplying.
4. 4(x 5)
5. 3x(x + 8)
(4 )x (4 )5
(3x )x + (3x )8
4x 20
3x2 + 24x
6. 2x(x 2 6x + 3)
(2x )x (2x )6x + (2x )3
2
2x3 12x2 + 6x
Multiply.
Multiply the polynomials.
4. −3x3(2x2 − 4x + 1)
Review for Mastery
7-8
To multiply monomials, multiply the constants, then multiply variables
with the same base.
Multiply 2x(x 2 + 3x + 7).
16. The length of a rectangle is 3 inches greater than the width.
a. Write a polynomial that represents the area
LESSON
8. 4x(x 2 + 8)
7. 5(x + 9)
5. (2x + 5) (9x2 + 6x)
18x 3 + 57x 2 + 30x
7. (2x3 + 6x + 8) (x2 − 5x + 1)
2x 5 − 10x 4 + 8x 3 − 22x 2 −
34x + 8
5x + 45
2
10. 3(5 x + 2)
2
3x 21
4x3 32x
3
11. (5a b) (2ab)
4
10a b
2
9. 3x 2 (2x 2 + 5x + 4)
6x4 + 15x3 + 12x2
12. 5y(y 2 + 7y 2)
5y3 + 35y2 10y
6/25/09 9:42:07 AM
84. Geometry Write a polynomial that represents the
volume of the rectangular prism. x 3 + 7x 2 + 10x
Because the last term
of the polynomial in
Exercise 87 is
negative, the signs of the second
terms in the binomials must be different. Eliminate choices A and B.
ÝÊÊÓ
85. Critical Thinking Is there any value for x that would
make the statement (x + 3)3 = x 3 + 3 3 true? Give an
Ý
ÝÊÊx
86. Estimation The length of a rectangle is 1 foot more than its width. Write a
polynomial that represents the area of the rectangle. Estimate the width of the
rectangle if its area is 25 square feet. x 2 + x ; 4.5 ft
Multiplying the first terms in
Exercise 88 gives 2a3, so choices F
and J cannot be correct. Students
who chose G probably did not distribute fully.
87. Which of the following products is equal to a 2 - 5a - 6?
(a - 1)(a - 5)
(a - 2)(a - 3)
(a + 2)(a - 3)
(a + 1)(a - 6)
88. Which of the following is equal to 2a (a 2 - 1)?
2a 2 - 2a
2a 3 - 1
2a 3 - 2a
2a 2 - 1
89. What is the degree of the product of 3x 3y 2z and x 2yz?
5
6
7
10
102–104. See p. A27.
Journal
Have students explain how to multiply (2x + 3) by (x + 4) using the
method of their choice.
CHALLENGE AND EXTEND
Simplify.
90. 6x
2
4x - 8
- 2(3x 2 - 2x + 4)
-x 2 - 6x
91. x - 2x (x + 3)
7x 2 + x
)
(
92. x 4x - 2 + 3x(x + 1)
2
ÝÊÊx
93. The diagram shows a sandbox and the frame that
surrounds it.
a. Write a polynomial that represents the area of the
sandbox. x 2 - 1
ÝÊÊ£
ÝÊÊÎ
ÝÊÊ£
Have students count off by fours.
Have the 1’s write a monomial, the
2’s write a binomial, the 3’s write a
trinomial, and the 4’s write a polynomial of degree four. Rearrange
the class into new groups of four, so
that each new group has a monomial, a binomial, a trinomial, and a
4th-degree polynomial. (Make some
groups of five students if needed.)
Have students multiply pairs of their
polynomials together. Each group
must turn in as many pairs as possible, with work shown for each
b. Write a polynomial that represents the area of the frame
that surrounds the sandbox. 8x + 16
94. Geometry The side length of a square is (8 + 2x) units. The area of this square is
the same as the perimeter of another square with a side length of (x 2 + 48) units.
Find the value of x. x = 4
95. Write a polynomial that represents the product of three consecutive integers.
Let x represent the first integer. x 3 + 3x 2 + 2x
96. Find m and n so that x m(x n + x n-2 )= x 5 + x 3. Possible answer: m = 2; n = 3
97. Find a so that 2x a(5x 2a-3 + 2x 2a+2) = 10x 3 + 4x 8 a = 2
SPIRAL REVIEW
98. A stop sign is 2.5 meters tall and casts a shadow that is 3.5 meters long. At the
same time, a flagpole casts a shadow that is 28 meters long. How tall is the
flagpole? (Lesson 2-8) 20 m
7-8
Find the distance, to the nearest hundredth, between each pair of points. (Lesson 5-5)
99. (2, 3) and (4, 6) 3.61
101. (-3, 7) and (-6, -2) 9.49
100. (-1, 4) and (0, 8) 4.12
Multiply.
Graph the solutions of each linear inequality. (Lesson 6-5)
102. y ≤ x - 2
103. 4x - 2y < 10
104. -y ≥ -3x + 1
7- 8 Multiplying Polynomials
LESSON
7-8
Problem Solving
7-8 PROBLEM SOLVING
Multiplying Polynomials
1. A bedroom has a length of x + 3 feet
and a width of x 1 feet. Write a
polynomial to express the area of the
bedroom. Then calculate the area if
x = 10.
2. The length of a classroom is 4 feet
longer than its width. Write a polynomial
to express the area of the classroom.
Then calculate the area if the width is
22 feet.
w2 + 4w
A1NL11S_c07_0512-0519.indd x
519
2
+ 2x 3
117 square feet
572 square feet
3. Nicholas is determining if he can afford
to buy a car. He multiplies the number
of months m by i + p + 30f where i
represents the monthly cost of
insurance, p represents the monthly car
payment, and f represents the number
of times he fills the gas tank each month.
Write the polynomial that Nicholas can
use to determine how much it will cost
him to own a car both for one month and
for one year.
i + p + 30f; 12i + 12p + 360f
4. A seat cushion is shaped like a
trapezoid. The shorter base of the
cushion is 3 inches greater than the
height. The longer base is 2 inches
shorter than twice the height. Write the
polynomial that can be used to find the
area of the cushion. (The area of a
trapezoid is represented by
1
h(b +b ) . )
2 1 2
3 h2 + 1 h
2
2
1
The volume of a pyramid can be found by using
Bh where B is the
3
area of the base and h is the height of the pyramid. The Great Pyramid
of Giza has a square base, and each side is about 300 feet longer than
the height of the pyramid. Select the best answer.
5. Which polynomial represents the
approximate area of the base of the
Great Pyramid?
A h + 90,000
B 2h + 90,000
C
h2 + 600h + 90,000
D 2h2 + 600h + 90,000
7. The original height of the Great Pyramid
was 485 feet. Due to erosion, it is now
about 450 feet. Find the approximate
volume of the Great Pyramid today.
A 562,500 ft3
C
B 616,225 ft3
D 99,623,042 ft3
84,375,000 ft3
LESSON
7-8
Challenge
7-8 CHALLENGE
The Missing Binomial
Determine the missing binomial. Choose from the table below.
Each binomial is used once.
x+4
x+1
x+1
x−3
x−4
x−6
x+3
x+6
x−2
+ 4) (x + 2) = x 2 + 2x + 4x + 8 = x 2 + 6x + 8
2. (x
+ 1) (x + 5) = x 2 + 5x + x + 5 + = x 2 + 6x + 5
3. (x
+ 2) (x − 3) = x 2 − 3x + 2x − 6 = x 2 − x − 6
4. (x
+ 3) (x − 5) = x 2 − 2x − 15
5. (x
+ 6) (x + 1) = x 2 + 7x + 6
6. (x
− 2) (x + 4) = x
+ 2x − 8
7. (x
− 3) (x + 6) = x 2 + 3x − 18
8. (x
− 4) (x + 2) = x 2 − 2x − 8
9. (x
− 6) (x − 5) = x 2 − 11x + 30
The binomials missing from the following equations are not all listed
in the table. Determine the missing binomials.
+ 2) (x + 7) = x 2 + 9x + 14
11. (x
+ 3) (x + 4) = x 2 + 7x + 12
12. (x
− 6) (x − 2) = x 2 − 8x + 12
13. (x
− 4) (x − 10) = x 2 − 14x + 40
14. (x
+ 6) (x + 3) = x 2 + 9x + 18
15. (x
+ 1) (x − 4) = x 2 − 3x − 4
F
1 3
h + 200h2 + 30,000h
3
16. (x
− 2) (x − 2) = x 2 − 4x + 4
17. (3x
− 1) (x + 2) = 3x 2 + 5x − 2
G
1 2
h + 200h + 30,000
3
18. (x
− 7) (2x + 3) = 2x 2 − 11x − 21
19. (2x
− 3) (3x + 4) = 6x 2 − x − 12
H h3 + 600h2 + 90,000h
2. 4xy2(x + y)
4x2y2 + 4xy3
3. (x + 2)(x - 8)
x2 - 6x - 16
4. (2x - 7)(x2 + 3x - 4)
2x3 - x2 - 29x + 28
5. 6mn(m2 + 10mn - 2)
6m3n + 60m2n2 - 12mn
6. (2x - 5y)(3x + y)
6x2 - 13xy - 5y2
10. (x
6. Which polynomial represents the
approximate volume of the Great
Pyramid?
18s3t3
7/20/09 5:13:33 PM
1. (x
2
519
1. (6s2t2)(3st)
7. A triangle has a base that is
4 cm longer than its height.
a. Write a polynomial that
represents the area of the
1
triangle. _ h2 + 2h
2
b. Find the area when the
height is 8 cm. 48 cm2
Also available on transparency
J 3h3 + 600h2 + 90,000h
Lesson 7-8
519
Organizer
Volume and Surface Area
Geometry
The volume V of a three-dimensional figure is the amount of space it
occupies. The surface area S is the total area of the two-dimensional
surfaces that make up the figure.
Geometry
Pacing:
1
Block __
day
2
Objective: Apply polynomial
GI
operations to finding areas of
geometric figures.
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V = πr h
S = 2π
πr 2 + 2πrh
π
2
£ ŰÜ
6Ú
Î
£
6ÊÚ
ûÀÀ Ó
Î
Remember
Example
Students review and apply volume
and surface area formulas for geometric figures.
Write and simplify a polynomial expression for the volume of the cone. Leave the
«£
1 πr 2h
V=_
3
1 π 6p 2 p + 1
=_
( )(
)
3
Close
Assess
The volume of any prism equals the
area of the base times the height.
Have students explain how to find
the volume of the triangular prism
in Problem 6. Find the area of the
base. It is 2y2 + y. Multiply this by
the height, which is 3y. The volume
is 6y3 + 3y2.
Choose the correct formula.
È«
Substitute 6p for r and p + 1 for h.
1 π 36p 2 p + 1
=_
( )(
)
3
Use the Power of a Product Property.
1 (36)π ⎡p 2 p + 1 ⎤
=_
)⎦
⎣ (
3
Use the Associative and Commutative Properties of Multiplication.
= 12πp 2(p + 1)
Distribute 12π p 2.
= 12πp 3 + 12πp 2
Try This
Write and simplify a polynomial expression for the volume of each figure.
2.
3.
1.
L Êx
2k 3 - k 2 - 6k
Ó Î
Ó
Î

£Ó
ÎL
L£
3b 3 - 12b 2 - 15b
 ÊÓ
Ó
4πn 3 - 16πn 2 + 16πn
Write and simplify a polynomial expression for the surface area of each figure.
4.
5.
ÓÝ
Ý ÊÎ
520
520
Chapter 7
16x + 30x + 6
Ü£
6.
19y 2 + 14y
Þ ÊÎ
ÜÎ
ÓÞ
ÎÞ
ÓÞ £
Chapter 7 Exponents and Polynomials
A1NL11S_c07_0520.indd 520
KEYWORD: MA7 Resources
ÓÝ Ê£
2
4πw 2 - 4π
6/25/09 9:43:46 AM
7-9
Special Products
of Binomials
7-9
Organizer
Block
__1 day
2
Objective: Find special products
of binomials.
Vocabulary
perfect-square trinomial
difference of two squares
GI
Why learn this?
You can use special products to find areas,
such as the area of a deck around a pond.
(See Example 4.)
Objective
Find special products of
binomials.
<D
@<I
Online Edition
Tutorial Videos
Imagine a square with sides of length (a + b):
Countdown Week 17
>ÊÊL
>
>
L
>Ó
>L
>L
Ó
>ÊÊL
Warm Up
L
L
Simplify.
1. 42
The area of this square is (a + b)(a + b), or (a + b) . The area of this square can
also be found by adding the areas of the smaller squares and rectangles inside.
The sum of the areas inside is a 2 + ab + ab + b 2.
2
3. (-2)2
2
2
49
4. (x)2
x2
2
m4
7. 2(6xy) 12xy 8. 2(8x2) 16x2
You can use the FOIL method to verify this:
F
4
5. -(5y2) -5y2 6. (m2)
This means that (a + b) = a + 2ab + b .
2
2. 72
16
Also available on transparency
L
(a + b)2 = (a + b)(a + b) = a 2 + ab + ab + b 2
I
O
= a 2 + 2ab + b 2
A trinomial of the form a 2 + 2ab + b 2 is called a perfect-square trinomial.
A perfect-square trinomial is a trinomial that is the result of squaring a
binomial.
EXAMPLE
Finding Products in the Form (a + b)
1
Parent: What happened in math
class today?
Student: When the teacher said to
look for perfect squares, everyone
looked at me.
2
Multiply.
A (x + 4)2
(a + b)2 = a 2 + 2ab + b 2
Use the rule for (a + b) .
(x + 4)2 = x 2 + 2(x)(4) + 4 2
Identify a and b: a = x and b = 4.
2
= x 2 + 8x + 16
B
Simplify.
(3x + 2y)2
(a + b)2 = a 2 + 2ab + b 2
Use the rule for (a + b) .
2
(3x + 2y)2 = (3x)2 + 2(3x)(2y) + (2y)2
= 9x 2 + 12xy + 4y 2
Identify a and b: a = 3x and b = 2y.
Simplify.
7- 9 Special Products of Binomials
521
1 Introduce
A1NL11S_c07_0521-0527.indd e
521
x p l o r at i o n
7-9
You can use the FOIL method to ﬁnd the square of a
binomial. For example, to ﬁnd a 1 2, ﬁrst write the
product as a 1 a 1 . Then use the FOIL method.
a 1 2 a 1 a 1 a a11a1
a2 a a 1
a 2 2a 1
2
2
1. Use the FOIL method to complete the table.
Power
a 1 2
a 2 6/25/09 9:46:19 AM
Motivate
Special Products
of Binomials
Expanded Form
Product
a 1 a 1 a 2 2a 1
2
a 3 2
a 4 2
2. Look for a pattern in the right column of the table. Use the
2
pattern to find a 5 without using the FOIL method.
Have students find the products of the following:
(x + 3)(x + 3)
(x + 4)(x + 4)
(x + 5)(x + 5)
x2 + 6x + 9
x2 + 8x + 16
x2 + 10x + 25
Discuss with students any patterns they see. Lead
them to recognize that the middle term of the
trinomial is two times the product of the first and
last terms of the binomial.
(a + b)(a + b) = a2 + 2ab + b2
3. Find a 9 2 without using the FOIL method.
THINK AND DISCUSS
4. Describe a general rule you can use to find a b 2.
5. Show how you can apply your rule to find 7 x 2.
Explorations and answers are provided in
Alternate Openers: Explorations Transparencies.
KEYWORD: MA7 Resources
Lesson 7-9
521
Multiply.
C
(4 + s 2)2
(a + b)2 = a 2 + 2ab + b 2
(4 + s 2)2 = (4)2 + 2(4)(s 2) + (s 2)2
Example 1
Use the rule for (a + b) .
2
Identify a and b: a = 4 and b = s 2.
= 16 + 8s 2 + s 4
Multiply.
A. (x + 3)2
D (-m + 3)2
x2 + 6x + 9
B. (4s + 3t)2
16s2 + 24st + 9t2
C. (5 +
25 +
)
m2 2
10m2
+
(a + b)2 = a 2 + 2ab + b 2
2
Identify a and b: a = -m and b = 3.
= m 2 - 6m + 9
Multiply.
1a. (x + 6)2
Multiply.
Simplify.
1b. (5a + b)
x + 12x + 36
25a + 10ab + b
1c. (1 + c 3)
2
2
1 + 2c + c 6
2
You can use the FOIL method to find products in the form (a - b) :
2
x2 - 12x + 36
B. (4m - 10)2 16m2 - 80m + 100
2
F
C. (2x - 5y)2 4x2 - 20xy + 25y2
D. (7 - r3)2
Use the rule for (a + b) .
(-m + 3)2 = (-m)2 + 2(-m)(3) + 3 2
m4
Example 2
A. (x - 6)2
Simplify.
2
3
L
(a - b)2 = (a - b)(a - b) = a 2 - ab - ab + b 2
49 - 14r3 + r6
I
O
= a 2 - 2ab + b 2
A trinomial of the form a - 2ab + b is also a perfect-square trinomial because
it is the result of squaring the binomial (a - b).
2
INTERVENTION
EXAMPLE
Questioning Strategies
EX A M P L E
2
2
Finding Products in the Form (a - b)
Multiply.
A (x - 5)2
1
(a - b)2 = a 2 - 2ab + b 2
(x - 5)2 = x 2 - 2(x)(5) + 5 2
• How do you find the middle term?
• In which position could a perfectsquare trinomial have a term with
an odd exponent?
EX A M P L E
2
Use the rule for (a - b) .
2
Identify a and b: a = x and b = 5.
= x 2 - 10x + 25
Simplify.
B (6a - 1)2
(a - b)2 = a 2 - 2ab + b 2
(6a - 1)2 = (6a)2 - 2(6a)(1) + (1)2
2
• How are these examples different
from those in Example 1? What
causes the differences? How are
they the same?
= 36 a 2 - 12a + 1
C (4c - 3d)
2
Identify a and b: a = 6a and b = 1.
Simplify.
2
(a - b)2 = a 2 - 2ab + b 2
(4c - 3d) 2 = (4c) 2 - 2(4c)(3d) + (3d)2
• Why is the last term of any perfectsquare trinomial always positive?
Use the rule for (a - b) .
= 16c 2 - 24cd + 9d 2
Use the rule for (a - b) .
2
Identify a and b: a = 4c and b = 3d.
Simplify.
D (3 - x )
2 2
(a - b)2 = (a)2 - 2ab + b 2
(3 - x 2)2 = (3) 2 - 2(3)(x 2) + (x 2)2
= 9 - 6x + x
2
2a. x 2 - 14x + 49
2b. 9b 2 - 12bc + 4c 2
2c. a 4 - 8a 2 + 16
522
Multiply.
2a. (x - 7)2
4
Use the rule for (a - b) .
2
Identify a and b: a = 3 and b = x 2.
Simplify.
2
2b. (3b - 2c)
2c. (a 2 - 4)
2
Chapter 7 Exponents and Polynomials
2 Teach
A1NL11S_c07_0521-0527.indd 522
6/25/09 9:46:42 AM
Guided Instruction
Review how to multiply any two binomials
before starting this section. Show the FOIL
2
2
method for (a + b) and (a - b) next to
each other, so students can better
visualize the similarities and differences.
Tell students that the binomials in this
lesson could be multiplied using any of the
methods they already know, but that
certain types of binomials can be multiplied more quickly knowing these rules.
522
Chapter 7
Through Auditory Cues
Have students learn the “verbal rules” for
the special products.
(a + b) 2 = a 2 + 2ab + b 2
(a - b) 2 = a 2 - 2ab + b 2
• first term squared
• plus (or minus) two times the product of
both terms
• plus last term squared
Have students create a similar verbal rule
for (a + b)(a - b) = a 2 - b 2.
You can use an area model to see that (a + b)(a - b) = a 2 - b 2.
1
2
>
L
>ÊÊL
>ÊÊL
L
>
L
>ÊÊL
Begin with a square
with area a 2. Remove
a square with area b 2.
The area of the new
figure is a 2 - b 2.
Then remove the
smaller rectangle on
the bottom. Turn it and
slide it up next to the
top rectangle.
Ê,,",
,/
>ÊÊL
L
>
3
>
""
The new arrangement
is a rectangle with
length a + b and
width a - b. Its area
is (a + b)(a - b).
When finding the product of
(a - b)2, some students might
believe that the middle term and
the last term will both be negative
numbers. Remind them that the last
term comes from a number being
squared, and thus will never be a
negative.
So (a + b)(a - b) = a 2 - b 2. A binomial of the form a 2 - b 2 is called a difference
of two squares .
EXAMPLE
3
Finding Products in the Form (a + b)(a - b)
Example 3
Multiply.
Multiply.
A (x + 6)(x - 6)
(a + b)(a - b) = a - b
2
2
(x + 6)(x - 6) = x 2 - 6 2
= x - 36
2
B
(x
2
Use the rule for (a + b)(a - b).
A. (x + 4)(x - 4)
Identify a and b: a = x and b = 6.
B. (p2 + 8q)(p2 - 8q) p4 - 64q2
Simplify.
+ 2y)(x - 2y)
C. (10 + b)(10 - b)
2
(a + b)(a - b) = a 2 - b 2
(x + 2y)(x 2 - 2y) = (x 2)2 - (2y)2
= x - 4y
Identify a and b: a = x 2 and b = 2y.
2
Simplify.
INTERVENTION
C (7 + n)(7 - n)
(a + b)(a - b) = a 2 - b 2
Use the rule for (a + b)(a - b).
(7 + n)(7 - n) = 7 2 - n 2
Identify a and b: a = 7 and b = n.
= 49 - n 2
Simplify.
Multiply.
3a. (x + 8)(x - 8) 3b. (3 + 2y 2)(3 - 2y 2)
x - 64
9 - 4y
2
EXAMPLE
4
100 - b2
Use the rule for (a + b)(a - b).
2
4
x2 - 16
4
Questioning Strategies
EX AM P LE
3
• Why is there no middle term in the
product (a + b)(a - b)?
• Why does the product
(a + b)(a - b) always have a
minus sign between the terms?
3c. (9 + r)(9 - r)
81 - r
2
Problem-Solving Application
A square koi pond is surrounded by a gravel
path. Write an expression that represents the
area of the path.
1
Understand the Problem
The answer will be an expression that represents
the area of the path.
List the important information:
• The pond is a square with a side length of
x - 2.
• The path has a side length of x + 2.
x–2
x+2
7- 9 Special Products of Binomials
A1NL11S_c07_0521-0527.indd 523
Math Background The binomials
(a + b) and (a - b) are called
conjugates. Conjugates are used in
many situations because of the special property that their product is always a difference
of squares.
523
6/25/09 9:47:01 AM
Lesson 7-9
523
2 Make a Plan
The area of the pond is (x - 2)2. The total area of the path plus the pond
is (x + 2)2. You can subtract the area of the pond from the total area to find
the area of the path.
Example 4
Write a polynomial that represents the area of the yard around
the pool shown below. 10x + 29
3 Solve
Step 1 Find the total area.
Use the rule for (a + b) : a = x and
b = 2.
(x + 2)2 = x 2 + 2(x)(2) + 2 2
ÝÊÊx
2
= x 2 + 4x + 4
ÝÊÊÓ
Step 2 Find the area of the pond.
ÝÊÊÓ
Use the rule for (a - b) : a = x and
b = 2.
2
(x - 2)2 = x 2 - 2(x)(2) + 2 2
ÝÊÊx
= x 2 - 4x + 4
To subtract a
opposite of each
term.
Step 3 Find the area of the path.
area of path =
a
INTERVENTION
Questioning Strategies
EX A M P L E
=
-
area of pond
x 2 + 4x + 4 -
( x 2 - 4x + 4)
total area
= x 2 + 4x + 4 - x 2 + 4x - 4
Identify like terms.
= (x 2 - x 2) + (4x + 4x) + (4 - 4) Group like terms together.
= 8x
4
• What are you finding when you are
multiplying the binomials?
The area of the path is 8x.
• What area are you asked to find?
Combine like terms.
4 Look Back
• What operation would you need to
perform to get this result?
Suppose that x = 10. Then one side of the path is 12, and the total
area is 12 2, or 144. Also, if x = 10, one side of the pond is 8, and the
area of the pond is 8 2, or 64. This means the area of the path is
144 - 64 = 80.
According to the solution above, the area of the path is 8x. If x = 10,
then 8x = 8(10) = 80. ✓
4. Write an expression that represents the
area of the swimming pool at right. 25
xÊÊÝ
xÊÊÝ
Ý
Ý
Special Products of Binomials
Perfect-Square Trinomials
(a + b) = (a + b)(a + b) = a 2 + 2ab + b 2
2
(a - b)2 = (a - b)(a - b) = a 2 - 2ab + b 2
Difference of Two Squares
(a + b)(a - b) = a 2 - b 2
524
Chapter 7 Exponents and Polynomials
3 Close
A1NL11S_c07_0521-0527.indd 524
Summarize
Have students state whether each product
is a perfect-square trinomial or a difference
of two squares, and then find the product.
(2x + 3y)2 perfect-square trinomial;
4x2 + 12xy + 9y2
(5m + 7) (5m - 7) difference of two squares;
25m2 - 49
(4s - t)2 perfect-square trinomial;
16s2 - 8st + t2
524
Tell students that recognizing special
products will greatly help them with
factoring in the next chapter.
Chapter 7
and INTERVENTION
Diagnose Before the Lesson
7-9 Warm Up, TE p. 521
Monitor During the Lesson
Check It Out! Exercises, SE pp. 522–524
Questioning Strategies, TE pp. 522–524
Assess After the Lesson
7-9 Lesson Quiz, TE p. 527
Alternative Assessment, TE p. 527
6/25/09 9:47:21 AM
1. (a + b)(a - b) =
THINK AND DISCUSS
a2 - ab + ab - b2 = a2 - b2
1. Use the FOIL method to verify that (a + b)(a - b) = a 2 - b 2.
2. product
2. When a binomial is squared, the middle term of the resulting trinomial
is twice the
?
of the first and last terms.
3. GET ORGANIZED Copy
and complete the graphic
organizer. Complete the
special product rules and
give an example of each.
7-9
3. See p. A7.
-«iV>Ê*À`ÕVÌÃÊvÊ>Ã
*iÀviVÌ-µÕ>Ài
/À>Ã
­>ÊÊL®ÓÊÊ¶
vviÀiViÊv
/ÜÊ-µÕ>ÀiÃ
­>ÊÊL®ÓÊÊ¶
­>ÊÊL®­>ÊÊL®ÊÊ¶
Exercises
7-9 Exercises
KEYWORD: MA11 7-9
KEYWORD: MA7 Parent
GUIDED PRACTICE
Assignment Guide
1. Vocabulary In your own words, describe a perfect-square trinomial.
Possible answer: a trinomial that is the result of squaring a binomial
Multiply.
SEE EXAMPLE
1
p. 521
SEE EXAMPLE
2
p. 522
SEE EXAMPLE
2. (x + 7)2
3. (2 + x)2
4. (x + 1)2
5. (2x + 6)2
6. (5x + 9)2
7.
8. (x - 6)2
9. (x - 2)2
11. (8 - x)2
3
p. 523
SEE EXAMPLE 4
p. 523
12.
17.
(2x
2
2
13. (7a - 2b)
15. (x + 6)(x - 6)
14. (x + 5)(x - 5)
+ 3)(2x - 3)
2
18.
16. (5x + 1)(5x - 1)
(9 - x )(9 + x )
3
If you finished Examples 1–2
Basic 21–32, 53–57
Average 21–32, 53–57
54, 56
(2a + 7b)2
10. (2x - 1) 2
(6p - q)2
3
19.
If you finished Examples 1–4
Basic 21–63, 64, 67–70,
75–82
Average 21–40, 42–52 even,
53–73, 75–82
(2x - 5y)(2x + 5y)
ÝÊÊÎ
20. Geometry Write a polynomial that represents
the area of the figure. 2x 2 + 8x + 10
ÝÊÊ£
ÝÊÊ£
ÝÊÊÎ
Assign Guided Practice exercises
as necessary.
Homework Quick Check
Quickly check key concepts.
Exercises: 22, 28, 36, 39, 42
PRACTICE AND PROBLEM SOLVING
Independent Practice
For
See
Exercises Example
21–26
27–32
33–38
39
1
2
3
4
Extra Practice
Skills Practice p. S17
Application Practice p. S34
Multiply.
21. (x + 3)2
24.
27.
(p + 2q 3)2
(s 2 - 7)2
22. (4 + z)2
23.
25. (2 + 3x)2
26.
28.
33. (a - 10)(a + 10)
34.
- 2)(x + 2)
37.
36.
(x
2
(r 2 + 5t)2
29. (a - 8)2
31. (3x - 4)2
30. (5 - w)2
2
(2c - d 3)2
(x 2 + y 2)2
32.
(y + 4)(y - 4)
(5a 2 + 9)(5a 2 - 9)
(1 - x 2)2
35. (7x + 3)(7x - 3)
38.
(x 3 + y 2)(x 3 - y 2)
7- 9 Special Products of Binomials
2. x2 + 14x + 49
3. 4 + 4x + x2
A1NL11S_c07_0521-0527.indd 525
+ 2x + 1
4.
x2
5.
4x2
+ 24x + 36
6. 25x2 + 90x + 81
14. x2 - 25
28. 4c2 - 4cd3 + d6
15. x2 - 36
29. a2 - 16a + 64
16.
25x2
17.
4x4
-1
30. 25 - 10w + w2
-9
31.
32. 1 - 2x2 + x4
19. 4x2 - 25y2
33. a2 - 100
x2
+ 6x + 9
21.
8. x2 - 12x + 36
22. 16 + 8z + z2
23.
x4
+
2x2y2
+
7/18/09 5:26:57 PM
- 24x + 16
18. 81 - x6
7. 4a2 + 28ab + 49b2
9. x2 - 4x + 4
9x2
525
34. y2 - 16
35. 49x2 - 9
y4
36. x4 - 4
10. 4x2 - 4x + 1
24. p2 + 4pq3 + 4q6
37. 25a4 - 81
11. 64 - 16x + x2
25. 4 + 12x +
38. x6 - y4
12. 36p2 - 12pq + q2
26. r 4 + 10r2t + 25t2
13. 49a2 - 28ab + 4b2
27. s4 - 14s2 + 49
9x2
KEYWORD: MA7 Resources
Lesson 7-9
525
39. πx 2 + 8πx +
16π
64a.
39. Entertainment Write a polynomial that
represents the area of the circular puzzle.
Remember that the formula for area of a circle
is A = πr 2, where r is the radius of the circle.
x
40a. x > 2;
40.
values less than or
equal to 2 cause the
width of the rectangle to be zero or
neg., which does not
make sense.
x
76.
x
Multi-Step A square has sides that are
(x - 1) units long and a rectangle has a length
of x units and a width of (x - 2) units.
r=x+4
a. What are the possible values of x ? Explain.
b. Which has the greater area, the square or
the rectangle? square
c. What is the difference in the areas? 1 sq. unit
Multiply.
77.
(x + y)2 x 2 + 2xy + y 2 42. (x - y)2x 2 - 2xy + y 2 43. (x 2 + 4)(x 2 - 4) x 4 - 16
2
2
44. (x 2 + 4) x 4 + 8x 2 + 16 45. (x 2 - 4) x 4 - 8x 2 + 16 46. (1 - x) 2 1 - 2x + x 2
49. x 6 - 2a 3x 3 + a 6 47. (1 + x)2 1 + 2x + x 2
48. (1 - x)(1 + x) 1 - x 2
49. (x 3 - a 3)(x 3 - a 3)
y
41.
50. (5 + n)(5 + n)
25 + 10n + n
(r - 4t 4)(r - 4t 4)
52.
r 2 - 8rt 4 + 16t 8
2
b
(a - b)2
a 2 - 2ab + b 2
1
4
(1 - 4) = 9
1 - 2(1)(4) + 4 2 = 9
53.
2
4
4
4
54.
3
2
1
1
a
b
(a + b)2
a 2 + 2ab + b 2
55.
1
4
25
25
56.
2
5
49
49
57.
3
0
9
9
a
b
(a + b)(a - b)
a2 - b2
58.
1
4
-15
-15
59.
2
3
-5
-5
60.
3
2
5
5
a
78.
36a - 25b
2
Copy and complete the tables to verify the special products of binomials.
x
51. (6a - 5b)(6a + 5b)
2
y
2
2
x
Math History
B.C.E., the Babylonians
lived in what is now
Iraq and Turkey. Around
575 B.C.E., they built
the Ishtar Gate to
serve as one of eight
main entrances into
the city of Babylon.
The image above is a
relief sculpture from a
restoration of the
Ishtar Gate.
7-9 PRACTICE A
61. Math History The Babylonians used tables of squares and the formula
(a + b)2 - (a - b)2
ab = _____________
to multiply two numbers. Use this formula to find the product
4
35 · 24. 840
62. Critical Thinking Find a value of c that makes 16 x 2 - 24x + c a perfect-square
trinomial. c = 9
63.
7-9 PRACTICE C
Practice B
LESSON
7-9
7-9 PRACTICE B
Special Products of Binomials
526
Multiply.
1. (x + 2)2
2. (m + 4)2
x 2 + 4x + 4
2
4. (2x + 5)
5. (3a + 2)
4x 2 + 20x + 25
9a 2 + 12a + 4
2
2
7. (b − 3)
8. (8 − y)
2
11. (4m − 9)
9x 2 − 42x + 49
2
6. (6 + 5b)
36 + 60b + 25b 2
9. (a − 10)
64 − 16y + y 2
2
10. (3x − 7)
Name ________________________________________ Date __________________ Class__________________
9 + 6a + a 2
2
b 2 − 6b + 9
16m 2 − 72m + 81
a 2 − 20a + 100
2
12. (6 − 3n)
36 − 36n + 9n 2
13. (x + 3) (x − 3)
14. (8 + y) (8 − y)
15. (x + 6) (x − 6)
x2 − 9
64 − y 2
x 2 − 36
16. (5x + 2) (5x − 2)
17. (10x + 7y) (10x − 7y)
25x 2 − 4
100x 2 − 49y 2
19. Write a simplified expression that represents the...
Explain the error below. What is the correct product?
(a - b) = a 2 - b 2 Possible answer: The square of a diff. is not the same as a diff.
of squares; a 2 - 2ab + b 2.
Chapter 7 Exponents and Polynomials
3. (3 + a)2
m 2 + 8m + 16
2
ANALYSIS/////
/////ERROR
2
18. (x2 + 3y) (x2 − 3y)
x 4 − 9y 2
LESSON
7-9
Use a Concept Map
When multiplying a binomial by another binomial, there are two special
kinds of products that may result. They are outlined in these concept
maps.
Definition
A1NL11S_c07_0521-0527.indd 526
LESSON
7-9
Review for Mastery
7-9 RETEACH
Special Products of Binomials
A perfect-square trinomial is a trinomial that is the result of squaring a binomial.
(a + b)2 = a2 + 2ab + b2
Add product of 2, a, and b.
(a − b)2 = a2 − 2ab + b2
Subtract product of 2, a, and b.
Examples
(a + b)2 = a2 + 2ab + b2
(n + 52) = n2 + 10n + 25
(a − b)2 = a2 − 2ab + b2
(3x − 2y)2 = 9xy2 − 12xy + 4y2
Multiply (x + 4)2.
(x + 4)2
2
x + 2(x)(4) + 4
4 − x2
2
x2 + 8x + 16
Formula
a: 4x
b: 3
32
16x2 − 2(4x)(3) + 32
Middle term is
subtracted.
16x2 − 24x + 9
Simplify.
Simplify.
(a + b) (a − b) = a2 − b2
c. area of the shaded area.
State whether each product will result in a perfect-square trinomial.
to the length and 2 units to the width.
a. What is the new area of the smaller rectangle?
16 − x 2
Examples
Difference of
Squares
Non Examples
(x + 4) (x + 4)
(x2 + y) (x2 − y) = x4 − y2
(y − 3) (y − 3)
1. Which type of special product is x2 − 64y2?
difference of squares
2. Which type of special product will result from
multiplying (k2 − 3) (k2 − 3)?
perfect square trinomial
4. (x + 7)2
Multiply. Then identify the type of special product.
4. (c2 + 10d) (c2 + 10d)
c 4 + 20c 2 d + 100d 2
perfect square trinomial
5. (2s + 3) (2s − 3)
4s 2 − 9
difference of squares
6. (2x + 10)2
2
Square a: x
Square a: x
14x
Square b: 49
2
x + 14x + 49
2x
Square b: 1
2
x − 2x + 1
2(a)(b):
2
Square a: x
40x
Square b: 100
2
4x + 40x + 100
2(a)(b):
Multiply.
x 2 − 16x + 64
It will have 3 terms.
yes
5. (x − 1)2
7. (x − 8)2
3. Why is (x + 4) (x + 4) not a difference of squares?
3. (5x − 6) (5x − 6)
Fill in the blanks. Then write the perfect-square trinomial.
2(a)(b):
2. (x + 2) (x − 2)
no
2
20
Chapter 7
1. (x + 5) (x + 5)
yes
(t + 6) (t − 6) = t2 − 36
b. What is the area of the new shaded area?
526
(4x − 3)2
b: 4
2
b. area of the small rectangle.
Multiply (4x − 3)2.
a: x
a. area of the large rectangle.
36 − x
7/20/09 2:56:08 PM
Square b.
Square a.
Perfect Square
Trinomial
Formulas
Square b.
Square a.
A trinomial (3 terms) that
is the result of squaring a
binomial (2 terms)
8. (x + 2)2
x 2 + 4x + 4
9. (7x − 5)2
49x 2 − 70x + 25
Exercise 64 involves
finding area using
polynomials. This
exercise prepares students for the
Multi-Step Test Prep on page 528.
64. This problem will prepare you for the Multi-Step Test Prep on page 528.
a. Michael is fencing part of his yard. He started with a square of length x on each
side. He then added 3 feet to the length and subtracted 3 feet from the width.
Make a sketch to show the fenced area with the length and width labeled.
b. Write a polynomial that represents the area of the fenced region. x 2 - 9
c. Michael bought a total of 48 feet of fencing. What is the area of his fenced region?
135 ft2
65. Critical Thinking The polynomial ax 2 - 49 is a difference of two squares. Find all
possible values of a between 1 and 100 inclusive. 1, 4, 9, 16, 25, 36, 49, 64, 81, 100
66. Write About It When is the product of two binomials also a binomial? Explain and
give an example. When 1 binomial is in the form a + b and the other is in the form
a - b ; (x + 2)(x - 2) = x 2 - 4
67. What is (5x - 6y)(5x - 6y)?
25x 2 - 22xy + 36y 2
25x 2 - 60xy + 36y 2
25x 2 + 22xy + 36y 2
25x 2 + 60xy + 36y 2
68. Which product is represented by the model?
(2x + 5)(2x + 5)
(5x - 2)(5x - 2)
(5x + 2)(5x - 2)
(5x + 2)(5x + 2)
69. If a + b = 12 and a 2 - b 2 = 96 what is the value of a?
2
4
8
10
70. If rs = 15 and (r + s)2 = 64, what is the value of r 2 + s 2?
25
30
34
49
ÓxÝÓ
£äÝ
£äÝ
{
3
71. Multiply (x + 4)(x + 4)(x - 4).
Have students describe three ways
to find the product (x + 2)(x + 2);
one method must use the special
products rule.
72. Multiply (x + 4)(x - 4)(x - 4).
2
73. If x + bx + c is a perfect-square trinomial, what is the relationship between b and c?
74. You can multiply two numbers by rewriting the numbers as the difference of two
squares. For example:
73. b = ±2 √c�
36 · 24 = (30 + 6)(30 - 6) = 30 2 - 6 2 = 900 - 36 = 864
Use this method to multiply 27 · 19. Explain how you rewrote the numbers.
513; rewrite 27 as 23 + 4 and 19 as 23 - 4.
SPIRAL REVIEW
75. The square paper that Yuki is using to make an origami frog has an area of 165 cm 2.
Find the side length of the paper to the nearest centimeter. (Lesson 1-5) 13 cm
79. 12x 2 + 6x
80. 4m + 6m +
2n + 11
4
For Exercise 69, a2 - b2 = 96 is the
difference of two squares, so
(a + b)(a - b) = 96, and by
substitution, 12(a - b) = 96. After
dividing both sides by 12, a - b = 8.
This equation and a + b = 12 form
a system of linear equations, easily
solved by elimination.
Journal
CHALLENGE AND EXTEND
71. x + 4x 2 16x - 64
72. x 3 - 4x 2 16x + 64
When multiplying
binomials of the form
(a - b)(a - b) in
Exercise 67, the middle term of the
product is negative. Choices C and D
can be eliminated.
3
81. -3p 3 - 8p
Have students create binomials of
2
2
the form (a + b) , (a - b) , and
(a + b)(a - b); find each product;
and describe how the special products rule is illustrated.
Use intercepts to graph the line described by each equation. (Lesson 5-2)
1x+y=4
76. 2x + 3y = 6
77. y = -3x + 9
78. _
2
7-9
Multiply.
79. 3x 2 + 8x - 2x + 9x 2
80.
82. 2t 2 + 16t + 17 81. (2p 3 + p) - (5p 3 + 9p)
82.
(8m 4 + 2n - 3m 3 + 6) + (9m 3 + 5 - 4m 4)
(12t - 3t 2 + 10) - (-5t 2 - 7 - 4t)
2
x2 + 14x + 49
2
x2 - 4x + 4
1. (x + 7)
2. (x - 2)
2
7- 9 Special Products of Binomials
Name _______________________________________ Date __________________ Class__________________
LESSON
7-9
7-9 PROBLEM SOLVING
Problem Solving
Special Products of Binomials
2. A museum set aside part of a large
gallery for a special exhibit.
1. This week Kyara worked x + 4 hours.
She is paid x 4 dollars per hour. Write
a polynomial for the amount that Kyara
earned this week. Then calculate her
pay if x = 12.
Name ________________________________________ Date __________________ Class__________________
7-9
In Exercises 1–5, use the given values of a and b to complete the
table below.
\$128
3. Gary is building a square table for a
kitchen. In his initial sketch, each side
measured x inches. After rearranging
some furniture, he realized he would
have to add one foot to the length and
remove one foot from the width and
have a rectangular table instead. Write
a polynomial to represent the area of
the rectangular table.
x2 144 in2
F 4x2 + 8
ab
36
81
121
9
4
9
9
169
−60
32
72
112
−160
−15
8
18
28
−40
−4
3
6
4
7
5
−8
0.75x 2 x 65
6.
5.
(a 2 + 2ab + b 2 ) − (a 2 − 2ab + b 2 ) = 4ab
H 4x2 + 16
ÝÊÊÇ
(a + b)2 + (a − b)2 = 2a 2 + 2b 2
8. [(a2 + b2) + (a2 − b2)]2
7. A 3-ft wide path is built around the
garden. Which expression represents the
area of the path?
F 12x + 33
24x + 84
H 4x2 + 28x + 29
J 4x2 + 40x + 100
2
B 0.86x + 3.44x + 28.56
C 7.14x2 + 28.56x + 3.44
D 7.14x2 + 3.44x + 28.56
Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.
7-73
ÝÊÊÈ
4ab
c. Use algebra to show that your response to part b is true.
of (a + b)2 + (a − b)2.
G
4x2 - 36xy + 81y2
7. Write a polynomial that represents the shaded area of the
figure below. 14x - 85
The difference (a + b)2 − (a − b)2 is four times the product
ab.
Write a conjecture for the simplification of each sum or difference.
Try to do as much of the simplification as you can mentally.
0.86x2 + 28.56x + 3.44
2
6. (m2 + 2n)(m2 - 2n) m4 - 4n2
a. From the table, how are the values of (a + b)2 − (a − b)2 and ab related?
to part a, write a simplification of (a + b)2 − (a + b)2.
x2 4x + 4
4x2 + 16x + 16 J 4x2 + 8x + 16
4. (2x - 9y)
7/18/09 5:07:17 PM
7. Using a table like the one that you completed in Exercises
1–5 and the values of a and b used there, write a simplification
6. Which polynomial represents the area
of the garden outside the fountain?
(Use 3.14 for .)
A
(a + b)2 − (a − b)2
64
−2
5. Which polynomial represents the area of
the garden, including the fountain?
G
(a − b)2
4
4.
4. Which polynomial represents the area of
the fountain?
D
(a + b)2
5
3.
2575 square feet
B x2 4x 4
b
−3
2.
A fountain is in the center of a square garden. The radius of the
fountain is x 2 feet. The length of the garden is 2x + 4 feet.
Use this information and the diagram for questions 4 7.
C x2 4
a
1.
Write a polynomial for the area of the
gallery that is not part of the exhibit.
Then calculate the area of that section
if x = 60.
3. (5x + 2y) 25x2 + 20xy + 4y2
5. (4x + 5y)(4x - 5y)
16x2 - 25y2
The expansions of (a + b)2, (a − b)2, and (a + b) (a − b) show patterns
involving a and b. You can discover other patterns involving a and
b when you combine these special products.
A1NL11S_c07_0521-0527.indd 527
x2 16
A 2x 4
7-9 CHALLENGE
Patterns in Special Products
Challenge
LESSON
527
Holt McDougal Algebra 1
ÝÊÊÇ
9. [(a2 + b2) − (a2 − b2)]2
4a 4
4b 4
Simplify each expression.
10. [(a2 + b2) + (a2 − b2)]2
ÝÊÊÈ
11. [(a2 + b2) − (a2 − b2)]2
[(a 2 + b 2 ) + (a 2 − b 2 )] 2
= (a 2 + a 2 + b 2 − b 2 ) 2
= (2a 2 ) 2 = 22a 4 = 4a 4
[(a 2 + b 2 ) − (a 2 − b 2 )] 2
= (a 2 − a 2 + b 2 + b 2 ) 2
= (2b 2 ) 2 = 22b 4 = 4b 4
Also available on transparency
Lesson 7-9
Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.
7-72
Holt McDougal Algebra 1
527
SECTION 7B
SECTION
7B
Polynomials
Don’t Fence Me In James has 500
feet of fencing to enclose a rectangular
region on his farm for some sheep.
Organizer
1. Make a sketch of three possible
regions that James could enclose
and give the corresponding areas.
Objective: Assess students’
GI
ability to apply concepts and skills
in Lessons 7-6 through 7-9 in a
real-world format.
<D
@<I
2. If the length of the region is x, find
an expression for the width. 250 - x
write an equation for the area of
the region. A = x (250 - x)
Online Edition
4. Graph your equation from Problem 3
on your calculator. Sketch the graph.
5. James wants his fenced region to
1. Possible answers: have the largest area possible using
Resources
Algebra I Assessments
A = 15,625 ft 2
www.mathtekstoolkit.org
Problem
Text Reference
1
Lesson 7-7
2
Lesson 7-6
3
Lesson 7-8
4
Lesson 7-9
5
Lesson 7-9
6
Lesson 7-9
FT
500 feet of fencing. Find this area
using the graph or a table of values. 15,625 ft 2
6. What are the length and width of the
region with the area from Problem
5? Describe this region. 125 ft × 125 ft; square
FT
A = 10,000 ft 2
FT
FT
A = 15,000 ft 2
FT
FT
4.
528
Chapter 7 Exponents and Polynomials
INTERVENTION
Scaffolding Questions
A1NL11S_c07_0528-0529.indd 528
1. What is the perimeter of a rectangle?
distance around the rectangle How is it
calculated? P = 2 + 2w
2. What are some possible values for and
w? Possible answers: = 40, w = 210;
= 30, w = 220; = 45, w = 205
3. What is the area of a rectangle? number
of square units inside the rectangle How
is it calculated? A = w
KEYWORD: MA7 Resources
528
Chapter 7
4. What is an appropriate viewing window?
Possible answer: 0 < x < 250
and 0 < y < 16,000
5. How can you find the greatest area from
the graph or table? Graph: find the ycoordinate of the highest point; Table:
find the greatest y-value.
6. How can you find the dimensions from
the graph or table? Graph: find the
x-coordinate of the highest point; Table:
look for the greatest y-value and find its
corresponding x-value.
Extension
Now James will use his fencing to create
three sides of a rectangular region and build
a wall for the fourth side. Now what is the
greatest possible area? Describe this region.
What is the minimum length of the wall?
31,250 ft2; 125 ft × 250 ft rectangle; 250 ft
6/25/09 9:49:40 AM
SECTION 7B
SECTION
1. 2r 6 + 4r 2 - 3r ; 2
2. -8y 3 + y 2 + 2y + 7; -8
3. t 4 - 12t 3 - 4t ; 1
Quiz for Lessons 7-6 Through 7-9
7-6 Polynomials
7B
Write each polynomial in standard form and give the leading coefficient.
1. 4r 2 + 2r 6 - 3r
2. y 2 + 7 - 8y 3 + 2y
4. n + 3 + 3n 3n + n + 3; 3 5. 2 + 3x
2
2
3
3x + 2; 3
6. -3a 2 + 16 + a 7 + a
Classify each polynomial according to its degree and number of terms.
7. 2x 3 + 5x - 4
a 7 - 3a 2 + a + 16; 1
Objective: Assess students’
mastery of concepts and skills in
Lessons 7-6 through 7-9.
9. 6p 2 + 3p - p 4 + 2p 3
8. 5b 2
10. x 2 + 12 - x
Organizer
3. -12t 3 - 4t + t 4
3
11. -2x 3 - 5 + x - 2x 7
12. 5 - 6b 2 + b - 4b 4
13. Business The function C(x) = x 3 - 15x + 14 gives the cost to manufacture x units
of a product. What is the cost to manufacture 900 units? \$728,986,514
Resources
Assessment Resources
Section 7B Quiz
14.
16.
18.
(10m + 4m ) + (7m + 3m)
(12d 6 - 3d 2 ) + (2d 4 + 1)
(7n 2 - 3n) - (5n 2 + 5n)
3
2
15. (3t - 2t) + (9t + 4t - 6)
2
2
2
17. (6y 3 + 4y 2 ) - (2y 2 + 3y)
Test & Practice Generator
19. (b - 10) - (-5b + 4b)
2
3
20. Geometry The measures of the sides of a triangle are shown
as polynomials. Write a simplified polynomial to represent the
perimeter of the triangle. 2s 3 + 4s 2 + 5s + 5
ÓÃÎÊÊ{
INTERVENTION
xÃ
Resources
{ÃÓÊÊ£
Intervention and
Enrichment Worksheets
7-8 Multiplying Polynomials
Multiply.
21. 2h 3 · 5h 5
22.
(s 8 t 4 )(-6st 3 )
23. 2ab (5a 3 + 3a 2 b)
(3k + 5)2
25.
(2x 3 + 3y)(4x 2 + y)
26.
24.
(p 2 + 3p)(9p 2 - 6p - 5)
27. Geometry Write a simplified polynomial expression for the area of a parallelogram
whose base is (x + 7) units and whose height is (x - 3) units. x 2 + 4x - 21 square units
(
)
7-9 Special Products of Binomials
Multiply.
(d + 9)2
2
32. (a - b)
28.
(2x + 5y)2
29. (3 + 2t)2
30.
33. (3w - 1)2
34. (c + 2)(c - 2)
31. (m - 4)2
35. (5r + 6)(5r - 6)
36. Sports A child’s basketball has a radius of (x - 5) inches. Write a polynomial that
represents the surface area of the basketball. (The formula for the surface area of a
sphere is S = 4πr 2 , where r represents the radius of the sphere.) Leave the symbol π
(
)
7–12, 14–19. See p. A27.
NO
A1NL11S_c07_0528-0529.indd 529
Intervention
TO
Worksheets
21–26, 28–35. See pp. A27–A28.
YES
Diagnose and Prescribe
INTERVENE
529
ENRICH
6/25/09 9:50:08 AM
GO ON? Intervention, Section 7B
CD-ROM
Lesson 7-6
7-6 Intervention
Activity 7-6
Lesson 7-7
7-7 Intervention
Activity 7-7
Lesson 7-8
7-8 Intervention
Activity 7-8
Lesson 7-9
7-9 Intervention
Activity 7-9
Online
Diagnose and
Prescribe Online
Enrichment, Section 7B
Worksheets
CD-ROM
Online
529
CHAPTER
Study Guide:
Review
7
Organizer
Vocabulary
index . . . . . . . . . . . . . . . . . . . . . . 488
quadratic . . . . . . . . . . . . . . . . . . 497
cubic . . . . . . . . . . . . . . . . . . . . . . 497
leading coefficient . . . . . . . . . . 497
scientific notation . . . . . . . . . . 467
organize and review key concepts
and skills presented in Chapter 7.
degree of a monomial . . . . . . . 496
monomial . . . . . . . . . . . . . . . . . . 496
degree of a polynomial . . . . . . 496
perfect-square trinomial . . . . 521
standard form of a
polynomial . . . . . . . . . . . . . . 497
difference of two squares . . . . 523
polynomial . . . . . . . . . . . . . . . . . 496
trinomial. . . . . . . . . . . . . . . . . . . 497
GI
binomial . . . . . . . . . . . . . . . . . . . 497
Objective: Help students
<D
@<I
Online Edition
Multilingual Glossary
Complete the sentences below with vocabulary words from the list above.
1. A(n)
?
polynomial is a polynomial of degree 3.
−−−−−−
2. When a polynomial is written with the terms in order from highest to lowest degree,
it is in
?
.
−−−−−−
3. A(n)
?
is a number, a variable, or a product of numbers and variables with
−−−−−−
whole-number exponents.
Countdown Week 17
Resources
4. A(n)
?
is a polynomial with three terms.
−−−−−−
5.
?
is a method of writing numbers that are very large or very small.
−−−−−−
PuzzleView
Test & Practice Generator
Multilingual Glossary Online
7-1 Integer Exponents (pp. 460– 465)
KEYWORD: MA7 Glossary
EXAMPLES
6. The diameter of a certain bearing is 2 -5 in.
Simplify this expression.
Simplify.
Lesson Tutorial Videos
CD-ROM
■
-2 -4
1 = -_
1
1
-2 -4 = - _
= -_
2·2·2·2
16
24
■
1. cubic
30
30 = 1
2. standard form of a polynomial
■
3. monomial
4. trinomial
5. scientific notation
1
6. _ in.
32
7. 1
8. 1
1
9. _
125
1
10. _, or 0.0001
10,000
1
11. _
16
1
12. _
256
27
13. _
4
1
_
14. 2
m
15. b
1
16. - _
2x2y4
17. 2b6c4
3a2
18. _
4c2
s3
19. _2
qr
530
Chapter 7
■
Simplify
Simplify.
7. (3.6) 0
9. 5
Any nonzero number raised to
the zero power is 1.
Evaluate r 3s -4 for r = -3 and s = 2.
r 3s -4
(-3)(-3)(-3)
27
(-3) 3(2) -4 = __ = - _
16
2·2·2·2
-3 4
a b
_
.
-3 4
c -2
4 2
b c
a b =_
_
c -2
a3
530
EXERCISES
-3
8. (-1) -4
10. 10 -4
Evaluate each expression for the given value(s) of
the variable(s).
11. b -4 for b = 2
( )
2b
12. _
5
-4
for b = 10
13. -2p 3q -3 for p = 3 and q = -2
Simplify.
14. m -2
15. bc 0
1 x -2y -4
16. - _
2
2b 6
17. _
c -4
3a 2c -2
18. _
4b 0
q -1r -2
19. _
s -3
Chapter 7 Exponents and Polynomials
A1NL11S_c07_0530-0535.indd 530
6/25/09 2:50:06 PM
7-2 Powers of 10 and Scientific Notation (pp. 466– 471)
■
Write 1,000,000 as a power of 10.
1,000,000
The decimal point is 6 places
1,000,000 = 10 6
to the right of 1.
Find the value of 386.21 × 10 5.
386. 2 1 0 0 0
Move the decimal point 5
places to the right.
38,621,000
■
21. 0.00001
EXERCISES
EXAMPLES
■
20. 10,000,000
Write 0.000000041 in scientific notation.
0.0 0 0 0 0 0 0 4 1 Move the decimal
point 8 places to the
right to get a number
between 1 and 10.
4.1 × 10 -8
22. 102
Find the value of each power of 10.
20. 10 7
21. 10 -5
23. 10-11
24. 325,000
Write each number as a power of 10.
22. 100
23. 0.00000000001
25. 1800
Find the value of each expression.
24. 3.25 × 10 5
25. 0.18 × 10 4
27. 0.000299
26. 17 × 10 -2
26. 0.17
28. 5.8 × 10-7, 6.3 × 10-3,
2.2 × 102, 1.2 × 104
27. 299 × 10 -6
29. \$38,500,000,000
28. Order the list of numbers from least to greatest.
6.3 × 10 -3, 1.2 × 10 4, 5.8 × 10 -7, 2.2 × 10 2
30. 59
31. 23 · 34
29. In 2003, the average daily value of shares traded
on the New York Stock Exchange was about
\$3.85 × 10 10. Write this amount in standard form.
32. b10
33. r5
34. x12
Simplify.
30. 5 3 · 5 6
31. 2 6 · 3 · 2 -3 · 3 3
The powers have the
same base.
32. b · b
33. r 4 · r
35. 1
1
, or
36. _
23
1
37. _
, or
54
1
38. _6
16b
39. g12h8
34. (x 3)
35. (s 3)
40. x4y2
7-3 Multiplication Properties of Exponents (pp. 474– 480)
EXERCISES
EXAMPLES
Simplify.
■
■
5 3 · 5 -2
5 3 · 5 -2
5 3 +(-2)
51
5
-3
36. (2
(a 4 · a -2) · (b -3 · b)
a 2 · b -2
a2
_
b2
■
(a -3b 2) -2
(a -3) -2 · (b 2) -2
Use properties to group
factors.
with the same base.
Write with a positive
exponent.
Power of a Product Property
a 6 · b -4
Power of a Power Property
a6
_
b4
Write with a positive
exponent.
8
4
37. (5
)
3 -1
38. (4b 3)
-2
a ·b ·b·a
a 4 · b -3 · b · a -2
4
2
41. -x4y2
)
2 -2
39. (g 3h 2)
-2
40. (-x y)
2
0
2
42. (x 2y 3)(xy
43. j6k9
1
44. _
5
45. m8n30
2
43. (j 2k 3)(j 4k 6)
)
3 4
44. (5 3 · 5 -2)
42. x6y15
4
41. -(x y)
2
45. (mn 3) (mn 5)
-1
5
46. 8 × 1011
3
46. (4 × 10 8)(2 × 10 3)
47. (3 × 10 2)(3 × 10 5)
48. (5 × 10
3
49. (7 × 10
5
50. (3 × 10
-4
51. (3 × 10
-8
)(2 × 10 )
6
)(2 × 10 )
5
47. 9 × 107
48. 1 × 1010
)(4 × 10 )
9
49. 2.8 × 1015
)(6 × 10 )
50. 6 × 101
-1
51. 1.8 × 10-8
52. In 2003, Wyoming’s population was about
5.0 × 10 5. California’s population was about
7.1 × 10 times as large as Wyoming’s. What was
the approximate population of California? Write
Study Guide: Review
A1NL11S_c07_0530-0535.indd 531
1
_
8
1
_
625
52. 3.55 × 107
531
6/25/09 7:07:09 PM
Study Guide: Review
531
7-4 Division Properties of Exponents (pp. 481– 487)
53. 64
54. m5
7
55. _
32
56. 6b
EXERCISES
EXAMPLES
x
Simplify _.
9
■
Simplify.
x2
x 9 = x 9-2 = x 7
_
x2
57. t3v4
Subtract the exponents.
58. 16
59. 5 ×
101
60. 2.5 × 107
28
53. _
22
m6
54. _
m
26 · 4 · 73
55. _
25 · 44 · 72
24b 6
56. _
4b 5
t 4v 5
57. _
tv
1
58. _
2
()
-4
61. 9
Simplify each quotient and write the answer in
scientific notation.
62. 7
59. (2.5 × 10 8) ÷ (0.5 × 10 7)
63. 16
60. (2 × 10 10) ÷ (8 × 10 2)
64. 8
65. z 2
7-5 Rational Exponents (pp. 488– 493)
66. 5x 2
4 3
67. x y
■
69. 0
6 12
Simplify each expression.
1
_
r 6 s 12 3
√r s = (
6 12
)
1
_
Definition of
1
_
= (r 6) 3 · (s 12) 3
71. 6
(
72. 1
73.
Simplify √r s .
3
3
70. 4
3n2
EXERCISES
EXAMPLES
68. m 2n 4
= r
+ 2n - 4; 3
) · (s )
1
6·_
3
74. -a6 - a4 + 3a3 + 2a; -1
= (r 2) · (s 4)
75. linear binomial
=r s
1
12 · _
3
Power of a Product
Property
Power of a Power
Property
Simplify exponents.
61. 81 2
62. 343 3
2
_
64. (2 6) 2
Simplify each expression. All variables represent
nonnegative numbers.
5
65. √
z 10
67.
76. quintic monomial
1
_
63. 64 3
2 4
77. quartic trinomial
1
_
1
_
1
_
bn
3
66. √
125 x 6
√
x 8y 6
3
68. √
m 6n 12
7-6 Polynomials (pp. 496– 501)
78. constant monomial
EXERCISES
EXAMPLES
■
■
Find the degree of the polynomial 3x + 8x .
3x 2 + 8x 5
8x 5 has the highest degree.
Find the degree of each polynomial.
69. 5
70. 8st 3
The degree is 5.
71. 3z 6
Classify the polynomial y 3 - 2y according to
its degree and number of terms.
Degree: 3
Terms: 2
Write each polynomial in standard form. Then give
73. 2n - 4 + 3n 2
74. 2a - a 4 - a 6 + 3a 3
The polynomial y 3 - 2y is a cubic binomial.
Classify each polynomial according to its degree and
number of terms.
2
5
72. 6h
75. 2s - 6
76. -8p 5
77. -m - m - 1
4
532
Chapter 7
78. 2
Chapter 7 Exponents and Polynomials
A1NL11S_c07_0530-0535.indd 532
532
2
7/20/09 5:17:24 PM
7-7 Adding and Subtracting Polynomials (pp. 504– 509)
(h 3 - 2h) + (3h 2 + 4h) - 2h 3
81. 3h3 - 3h2 + 5
79. 3t + 5 - 7t - 2
82. 2m2 - 5m - 1
83. p2 + 5p + 8
(h 3 - 2h) + (3h 2 + 4h) - 2h 3
(h 3 - 2h 3) + (3h 2) + (4h - 2h)
80. 4x 5 - 6x 6 + 2x 5 - 7x 5
-h 3 + 3h 2 + 2h
82. (3m - 7) + (2m 2 - 8m + 6)
(n 3 + 5 - 6n 2) - (3n 2 - 7)
(n + 5 - 6n ) + (-3n + 7)
(n 3 + 5 - 6n 2) + (-3n 2 + 7)
n 3 + (-6n 2 - 3n 2) + (5 + 7)
3
84. -7z2 - z + 10
81. -h 3 - 2h 2 + 4h 3 - h 2 + 5
85. 3g2 + 2g + 4
86. -x2 + 4x + 8
83. (12 + 6p) - (p - p + 4)
2
Subtract.
■
80. -6x6 - x5
EXERCISES
EXAMPLES
■
79. -4t + 3
2
2
87. 8r2
84. (3z - 9z 2 + 2) + (2z 2 - 4z + 8)
88. 6a6b
85. (10g - g 2 + 3) - (-4g 2 + 8g - 1)
89. 18x3y2
92.
90. 3s6t14
(-5x 3 + 2x 2 - x + 5) - (-5x 3 + 3x 2 - 5x - 3)
91. 2x2 - 8x + 12
n 3 - 9n 2 + 12
92. -3a2b2 + 6a3b2 - 15a2b
93. a2 - 3a - 18
94. b2 - 6b - 27
7-8 Multiplying Polynomials (pp. 512– 519)
EXAMPLES
Multiply.
■ (2x
- 4)(3x + 5)
2x(3x) + 2x(5) - 4(3x) - 4(5)
6x 2 + 10x - 12x - 20
6x 2 - 2x - 20
■ (b
- 2)(b 2 + 4b - 5)
b(b 2) + b(4b) - b(5) - 2(b 2) - 2(4b) - 2(-5)
b 3 + 4b 2 - 5b - 2b 2 - 8b + 10
b 3 + 2b 2 - 13b + 10
95. x2 - 12x + 20
EXERCISES
96. t2 - 1
97. 8q2 + 34q + 30
Multiply.
87. (2r)(4r)
88. (3a )(2ab)
89. (-3xy)(-6x 2y)
1 s 2t 8
90. (3s 3t 2)(2st 4) _
2
91. 2(x 2 - 4x + 6)
92. -3ab(ab - 2a 2b + 5a)
101. m2 + 12m + 36
93. (a + 3)(a - 6)
94. (b - 9)(b + 3)
102. 9c2 + 42c + 49
95. (x - 10)(x - 2)
96. (t - 1)(t + 1)
97. (2q + 6)(4q + 5)
98. (5g - 8)(4g - 1)
98. 20g2 - 37g + 8
5
(
)
99. p2 - 8p + 16
100. x2 + 24x + 144
103. 4r2 - 4r + 1
104. 9a2 - 6ab + b2
105. 4n2 - 20n + 25
106. h2 - 26h + 169
107. x2 - 1
7-9 Special Products of Binomials (pp. 521–527)
EXAMPLES
Multiply.
■
109. c4 - d 2
EXERCISES
110. 9k4 - 49
Multiply.
(2h - 6)2
(2h - 6) 2 = (2h) 2 + 2(2h)(-6) + (-6) 2
99. (p - 4) 2
100. (x + 12) 2
101. (m + 6)2
102. (3c + 7)2
4h - 24h + 36
103. (2r - 1) 2
2
104. (3a - b)
(4x - 3)(4x + 3)
(4x - 3)(4x + 3) = (4x) 2 - 3 2
105. (2n - 5)2
2
106. (h - 13)
107. (x - 1)(x + 1)
108. (z + 15)(z - 15)
16x 2 - 9
109. (c - d)(c + d)
110. (3k 2 + 7)(3k 2 - 7)
2
■
108. z2 - 225
2
2
Study Guide: Review
A1NL11S_c07_0530-0535.indd 533
533
7/18/09 5:04:22 PM
Study Guide: Review
533
CHAPTER
7
Organizer
Evaluate each expression for the given value(s) of the variable(s).
( )
1b
1. _
3
Objective: Assess students’
Simplify.
GI
mastery of concepts and skills in
Chapter 7.
@<I
<D
-2
1
_
for b = 12
2. (14 - a 0b 2)
16
_
_2
3. 2r -3
r
for a = -2 and b = 4 -
_
2
5. m 2n -3 m
4. -3f 0g -1 - 3
g
3
-3
n
8
t
_
3
1 s -5t 3
6. _
2
3
_1
2s 5
Write each number as a power of 10.
Online Edition
7. 0.0000001 10-7
8. 10,000,000,000,000 10 13
9. 1 10 0
Find the value of each expression.
Resources
10. 1.25 × 10 -5 0.0000125
Assessment Resources
11. 10 8 × 10 -11 0.001
12. 325 × 10 -2 3.25
Chapter 7 Tests
13. Technology In 2002, there were approximately 544,000,000 Internet users
worldwide. Write this number in scientific notation. 5.44 × 10 8
• Free Response
(Levels A, B, C)
Simplify.
14. (f
• Multiple Choice
(Levels A, B, C)
)
4 3
15. (4b 2)
f 12
0
16. (a 3b 6)
1
6
17. -(x 3) · (x 2)
5
a 18b 36
6
-x 27
Simplify each quotient and write the answer in scientific notation.
• Performance Assessment
18. (3.6 × 10 9) ÷ (6 × 10 4) 6 × 10 4
19. (3 × 10 12) ÷ (9.6 × 10 16) 3.125 × 10-5
Simplify.
Test & Practice Generator
_
y4 3
20. _
y y
_
d 2f 5
f9
21. _
2
(d 3) f -4 d 4
1
( ) · (_2s6t ) _
16
25 · 33 · 54 3
22. _
28 · 32 · 54 8
4s
23. _
3t
-2
2
24. Geometry The surface area of a cone is approximated by the polynomial
3.14r 2 + 3.14r, where r is the radius and is the slant height. Find the approximate
surface area of a cone when = 5 cm and r = 3 cm. 75.36 cm 2
Simplify each expression. All variables represent nonnegative numbers.
( ) _35
27
25. _
125
1
_
3
3
26. √
43 3 43
4
√
25y 8 5y
-10b 3 - 6b 2
29. 3a - 4b + 2a 5a - 4b
27.
30. (2b - 4b
2
) - (6b
3
3
+ 8b
2
)
Multiply.
32. -5(r s - 6) -5r s + 30
33. (2t - 7)(t + 4) 2t + t - 28
35. (m + 6) 2 m 2 + 12m + 36
36. (3t - 7)(3t + 7) 9t 2 - 49
2
2
2
5
28. √
3 5 t 10 3t 2
-4g 3 - 6g 2 + g - 4
31. -9g 2 + 3g - 4g 3 - 2g + 3g 2 - 4
16g 3 - 24g 2 - 7g + 3
34. (4g - 1)(4g 2 - 5g - 3)
37. (3x 2 - 7)
2
9x 4 - 42x 2 + 49
38. Carpentry Carpenters use a tool called a speed square to help them mark right
angles. A speed square is a right triangle.
a. Write a polynomial that represents the area of the speed square shown.
b. Find the area when x = 4.5 in. 3.75 in 2
534
534
Chapter 7
ÓÝÊÊÈ
Chapter 7 Exponents and Polynomials
A1NL11S_c07_0530-0535.indd 534
KEYWORD: MA7 Resources
x 2 - x - 12
ÝÊÊ{
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7/18/09 5:04:26 PM
CHAPTER
7
Organizer
FOCUS ON SAT
percentile for each score. The percentile tells you what
percent of students scored lower than you on the same
test. Your percentile at the national and state levels may
differ because of the different groups being compared.
You may want to time yourself as you take this practice test.
It should take you about 7 minutes to complete.
1. If (x + 1)(x + 4) - (x - 1)(x - 2) = 0, what is
the value of x?
Objective: Provide practice for
You may use some types of calculators
on the math section of the SAT. For
about 40% of the test items, a graphing
calculator is recommended. Bring a
calculator that you are comfortable
using. You won’t have time to figure out
how a new calculator works.
GI
college entrance exams such as the
SAT.
(A) -37
1
(B) - _
4
(B) -25
Questions on the SAT represent the
following math content areas:
Number and Operations, 20–25%
(D) 7
1
(D) _
4
Algebra and Functions, 35–40%
(E) 27
Geometry and Measurement,
25–30%
(E) 1
5
2. Which of the following is equal to 4 ?
Data Analysis, Statistics, and
Probability, 10–15%
5. What is the area of a rectangle with a length of
x - a and a width of x + b?
ÝÊÊ>
I. 3 5 × 1 5
(A) x 2 - a 2
II. 2 10
(B) x 2 + b 2
III. 4 0 × 4 5
(C) x 2 - abx + ab
ÝÊÊL
(A) I only
(D) x 2 - ax - bx - ab
(B) II only
(E) x 2 + bx - ax - ab
• Algebra and Functions
• Geometry and Measurement
Text References:
Item
Lesson
(C) I and II only
(D) II and III only
(E) I, II, and III
Online Edition
College Entrance Exam
Practice
(C) -5
(C) 0
@<I
Resources
4. What is the value of 2x 3 - 4x 2 + 3x + 1 when
x = -2?
(A) -1
<D
6. For integers greater than 0, define the following
operations.
1
7-8
2
7-1,
7-3
3
7-1
4
7-6
5
7-8
6
7-7
a □ b = 2a 2 + 3b
3. If x -4 = 81, then x =
(A) -3
1
(B) _
4
1
(C) _
3
(D) 3
(E) 9
a b = 5a 2 - 2b
What is (a □ b) + (a b)?
(A) 7a 2 + b
(B) -3a 2 + 5b
(C) 7a 2 - b
(D) 3a 2 - 5b
(E) -3a 2 - b
College Entrance Exam Practice
1. Students who chose A or E may have
made a sign error in the second term.
Suggest that students use parentheses
around each product of binomials until
the multiplication is complete and then
distribute the negative sign from the
subtraction of the terms.
A1NL11S_c07_0530-0535.indd 535
2. Students who chose A, C, or E may
have added the bases in choice I and
kept the exponent the same. Remind
students that if the bases are the same
when powers are multiplied, then they
3. Students who chose D found the value
of x if x4 = 81. Suggest that
students rewrite x-4 with a positive
exponent and try again.
4. Students who chose C probably made
a sign error in the first or second term.
Remind students to be careful when
using exponents with negative numbers. Be sure they understand when
the negative sign is part of the base,
and when it is not.
535
5. Students who did not choose E should
review multiplying binomials. Remind
7/18/09 5:04:40 PM
students to be cautious of signs.
6. Students who chose B subtracted the
Remind students to read each test item
carefully.
College Entrance Exam Practice
535
CHAPTER
7
Organizer
Any Question Type: Use a Diagram
Objective: Provide opportunities
When a test item includes a diagram, use it to help solve the problem. Gather
as much information from the drawing as possible. However, keep in mind that
diagrams are not always drawn to scale and can be misleading.
GI
to learn and practice common testtaking strategies.
<D
@<I
Online Edition
Multiple Choice What is the height of the triangle when x = 4 and y = 1?
This Test Tackler
explains how test
item diagrams can
students assume that visual information from a diagram is correct, they
will likely misinterpret the information. Advise students that diagrams
are not always drawn to scale.
Students should not rely solely on
the appearance of the drawing to
should look closely at a drawing’s
labels. They may even need to
redraw the diagram to scale in order
to better represent the problem.
Explain that even though a test item
may not include a diagram, it may
be beneficial for students to make
a quick sketch. Show students the
importance of labeling their sketch
with the information in the test item.
2
8
4
16
­ÝÞ®Ó
In the diagram, the height appears to be less than 6, so you might
eliminate choices C and D. However, doing the math shows that
the height is actually greater than 6. Do not rely solely on visual
information. Always use the numbers given in the problem.
È
The height of the triangle is (xy)2.
When x = 4 and y = 1, (xy) 2 = (4 · 1) 2 = (4) 2 = 16.
Choice D is the correct answer.
If a test item does not have a diagram, draw a quick sketch of the problem situation. Label
your diagram with the data given in the problem.
Short Response A square placemat is lying in the middle of a rectangular table.
x
The side length of the placemat is __
. The length of the table is 12x, and the
2
()
width is 8x. Write a polynomial to represent the area of the placemat. Then write a
polynomial to represent the area of the table that surrounds the placemat.
£ÓÝ
Use the information in the problem to draw and label a diagram.
Then write the polynomials.
area of placemat = s 2 =
x
(_2x ) = (_2x )(_2x ) = _
4
2
area of table = w = (12x)(8x) = 96x 2
area of table - area of placemat = 96x 2 -
Ý
Ú
Ó
nÝ
2
Ú
ÊÝÊ
Ó
x
384x - x = _
383x
_
=_
2
2
4
4
2
2
4
2
x
.
The area of the placemat is __
4
2
383x
The area of the table that surrounds the placemat is _____
.
4
536
Chapter 7 Exponents and Polynomials
A1NL11S_c07_0536-0537.indd Sec1:536
536
Chapter 7
6/25/09 9:52:13 AM
If a given diagram does not reflect the problem,
draw a sketch that is more accurate. If a test
item does not have a diagram, use the given
information to sketch your own. Try to make your
sketch as accurate as possible.
Item C
Short Response Write a polynomial expression
1. the width
for the area of triangle QRP. Write a polynomial
expression for the area of triangle MNP. Then
use these expressions to write a polynomial
expression for the area of QRNM.
that follow.
a rectangle and labeling each
dimension, you will have a better
understanding of what should be
substituted into the area formula
for length and width.
+
Item A
Short Response The width of a rectangle is
£ä
3.
Ý
1.5 feet more than 4 times its length. Write
a polynomial expression for the area of the
rectangle. What is the area when the length is
16.75 feet?
*
£ÊÊÝ
x
,
È
x
7. Describe how redrawing the figure can help
you better understand the information in
the problem.
1. What is the unknown measure in this
problem?
the problem?
4. greater
5. No; it appears that the length of
rectangle ABDC is less than the
length of rectangle MNPO, and
this is not consistent with the
dimensions that are given.
8. After reading this test item, a student redrew
the figure as shown below. Is this a correct
interpretation of the original figure? Explain.
3. Draw and label a sketch of the situation.
A
Item B
Multiple Choice Rectangle ABDC is similar to
rectangle MNPO. If the width of rectangle ABDC
is 8, what is its length?
D
C
£ÓÝ
M
{
n
B
"
*
2
2x
24x
24
4. Look at the dimensions in the diagram.
Do you think that the length of rectangle
ABDC is greater or less than the length of
rectangle MNPO?
5. Do you think the drawings reflect the
information in the problem accurately? Why
or why not?
Item D
Multiple Choice The measure of angle XYZ
O
is (x 2 + 10x + 15)°. What is the measure of
angle XYW ?
(6x + 15)°
(2x 2 + 14x + 15)°
(6x 2 + 15)°
8
­ÝÓÊÊ{Ý®Â
9
N
<
8. No, it is not correct. The student
mislabeled the base of triangle
QRP. It should be labeled 7 + x.
9. What information does the diagram provide
that the problem does not?
9. the measure of angle WYZ
10. Will the measure of angle XYW be less than
or greater than the measure of angle XYZ ?
Explain.
10. Less than; angle XYW lies within
angle XYZ, so angle XYW must be
smaller.
6. Draw your own sketch to match the
information in the problem.
Test Tackler
P
the figures into 2 separate
triangles, you can better see the
base and height measures, which
area formula for each triangle.
7
(14x + 15)°
x
537
A. 4x2 + 1.5x; 1147.375 ft2
B. C
A1NL11S_c07_0536-0537.indd Sec1:537
6/25/09 9:52:35 AM
9
1
1
C. 5x + 35; _(x2 + x); -_ x2 + _x + 35
2
2
2
D. F
KEYWORD: MA7 Resources
Test Tackler
537
CHAPTER
7
KEYWORD: MA7 TestPrep
Organizer
CUMULATIVE ASSESSMENT, CHAPTERS 1–7
Objective: Provide review
GI
and practice for Chapters 1–7 and
standardized tests.
<D
@<I
Multiple Choice
6. A restaurant claims to have served 352 × 10 6
hamburgers. What is this number in scientific
notation?
1. A negative number is raised to a power. The
result is a negative number. What do you know
Online Edition
Resources
It is an even number.
3.52 × 10 8
It is an odd number.
3.52 × 10 4
It is zero.
352 × 10 6
It is a whole number.
Assessment Resources
3.52 × 10 6
7. Janet is ordering game cartridges from an online
retailer. The retailer’s prices, including shipping
and handling, are given in the table below.
2. Which expression represents the phrase eight less
than the product of a number and two?
Chapter 7 Cumulative Test
KEYWORD: MA7 TestPrep
2 - 8x
Game Cartridges
8 - 2x
1
54.95
2x - 8
2
104.95
3
154.95
4
204.95
x -8
_
2
3. An Internet service provider charges a \$20 set-up
fee plus \$12 per month. A competitor charges
\$15 per month. Which equation can you use to
find x, the number of months when the total
charge will be the same for both companies?
Total Cost (\$)
Which equation best describes the relationship
between the total cost c and the number of game
cartridges g?
c = 54.95g
15 = 20 + 12x
c = 51g + 0.95
20 + 12x = 15x
c = 50g + 4.95
20x + 12 = 15x
c = 51.65g
20 = 15x + 12x
8. Which equation describes a line parallel
to y = 5 - 2x?
4. Which is a solution of the inequality
7 - 3(x - 3) > 2(x + 3)?
y = -2x + 8
1x
y=5+_
2
y = 2x - 5
1x
y=5-_
2
0
2
5
12
9. A square has sides of length x - 4. A rectangle
has a length of x + 2 and a width of 2x - 1. What
is the total combined area of the square and the
rectangle?
5. One dose of Ted’s medication contains
0.625 milligram, or _58_ milligram, of a drug. Which
expression is equivalent to 0.625?
10x - 14
4x - 3
5(4) -2
3x 2 - 5x + 14
5(2) -4
3x 2 + 3x - 18
5(-2) 3
5(2) -3
538
Chapter 7 Exponents and Polynomials
1. B
14. 2
2. H
15. 4
A11NLS_c07_0538-0539.indd 538
3. B
16. 16
4. F
17a. \$28.00
5. D
b. \$19.60
6. G
7. C
8. F
9. C
10. J
KEYWORD: MA7 Resources
538
Chapter 7
3
, or 0.75
13. __
4
11. C
12. H
12/14/09 11:23:04 A
A
Test writers develop multiple-choice test options
with distracters. Distracters are incorrect options
that are based on common student errors. Be
cautious! Even if the answer you calculated is one
of the options, it may not be the correct answer.
10. Jennifer has a pocketful of change, all in nickels
and quarters. There are 11 coins with a total
value of \$1.15. Which system of equations can
you use to find the number of each type of coin?
⎧ n + q = 11
⎨
⎩ n + q = 1.15
Short Response
a. Find the price of the sweater while on the sale
1 Point = The student’s answer contains attributes of an appropriate
response but is flawed.
b. Find the price of the sweater while on the
0 Points = The student’s answer
contains no attributes of an appropriate response.
a. Find a 2, b 2, and c 2 when a = 2x, b = x 2 - 1, and
c = x 2 + 1. Show your work.
b. Is (2x, x 2 - 1, x 2 + 1) a Pythagorean triple?
Extended-Response
Rubric
19. Ron is making an ice sculpture. The block of ice is
Item 21
in the shape of a rectangular prism with a length
of (x + 2) inches, a width of (x - 2) inches, and a
height of 2x inches.
⎧ n + q = 11
⎨
⎩ 0.05n + 0.25q = 1.15
4 Points = The student writes the
correct expression with full work
or explanation in part a, shows
one method of checking the previous answer in part b, writes the
correct area as a binomial square
in part c, expands the product correctly and identifies the type of
polynomial in part d, and answers
correctly with explanation in
part e.
a. Write and simplify a polynomial expression for
11. Which of the following is a true statement?
the volume of the block of ice. Show your work.
p
⎡(a m) n ⎤ = a m+n+p
⎣
⎦
p
⎡(a m) n ⎤ = a mn+p
⎣
⎦
b. The final volume of the ice sculpture is
(x 3 + 4x 2 - 10x + 1) cubic inches. Write an
expression for the volume of ice that Ron
p
⎡(a m) n ⎤ = a mnp
⎣
⎦
p
⎡(a m) n ⎤ = (a m+n) p
⎣
⎦
20. Simplify the expression (3 · a 2 · b -4 · a · b -3)
12. In 1867, the United States purchased the Alaska
Territory from Russia for \$7.2 × 10 6. The total
area was about 6 × 10 5 square miles. What was
the price per square mile?
-3
using two different methods. Show that the
results are the same.
3 Points = The student writes the
correct expression with minimal
work or explanation in part a,
shows one method of checking the previous answer in part
b, writes the area as a binomial
square with minor errors in part c,
expands the product with minor
errors and identifies the type of
polynomial in part d, and answers
correctly with explanation in
part e.
Extended Response
21. Look at the pentagon below.
2x - 2
4x - 4
Gridded Response
4x - 4
13. Evaluate the expression 3b-2c 0 for b = 2 and c = -3.
a. Write and simplify an expression that
represents the area of the pentagon. Show
14. What is the slope of the line described by
-3y = -6x - 12?
2 Points = The student answers
parts a, b, and e correctly with
attempted explanation; or the student answers parts c, d, and e correctly with attempted explanation.
b. Show one method of checking that your
15. The quotient (5.6 × 10 ) ÷ (8 × 10 ) is written in
scientific notation as (7 × 10 n). What is the value
8
expression in part a is correct.
3
c. The triangular part of the pentagon can be
rearranged to form a square. Write the area of
this square as the square of a binomial.
of n?
d. Expand the product that you wrote in part c.
16. The volume of a plastic cylinder is 64 cubic
centimeters. A glass cylinder has the same height
and a radius that is half that of the plastic
cylinder. What is the volume in cubic centimeters
of the glass cylinder?
What type of polynomial is this?
e. Is the square of a binomial ever a binomial?
Cumulative Assessment, Chapters 1–7
-3
20. (3 · a2 · b-4 · a · b-3) =
18a.
2 Points = The student’s answer is
Pythagorean triple if a 2 + b 2 = c 2.
⎧ 5n + 25q = 11
⎨
⎩ n + q = 1.15
PM
A1NL11S_c07_0538-0539.indd 539
Items 17–20
down 20% and placed on the sale rack. Later, the
sweater was marked down an additional 30% and
placed on the clearance rack.
18. A set of positive integers (a, b, c) is called a
⎧ n + q = 11
⎨
⎩ 5n + 25q = 1.15
a2
Short-Response Rubric
17. A sweater that normally sells for \$35 was marked
= (2x) =
2
4x2
= (x2 - 1)2 = x4 2x2 + 1
b2
c2 = (x2 + 1)2 = x4 +
2x2 + 1
b. Yes; (2x)2 + (x2 - 1)2 =
4x2 + x4 - 2x2 + 1 =
x4 + 2x2 + 1 and
(x2 + 1)2 = x4 +
2x2 + 1.
Because (2x)2 +
(x2 - 1)2 = (x2 + 1)2,
the expressions form a
Pythagorean triple.
19a. (2x3 - 8x) in3
b. (x3 - 4x2 + 2x - 1) in3
(3 · a3 ·
-3
b-7
) =
3-3 · a-9 · b21 =
b21
_
;
27a9
(3 · a2 · b-4 · a · b-3)-3 =
3-3 · a-6 · b12 · a-3 · b9 =
3-3 · a-9 · b21 =
b21
_
27a9
21a. 20x2 - 40x + 20
Substitute 5 for x in the
diagram. Then the side
lengths of the square
are 16 units, and the
height of the triangle is
8 units. The area of the
539
square is 256 square
units, and the area of
the triangle is 64 square7/18/09
which gives the area
of the pentagon,
320 square units. Then
substitute 5 for x in the
simplified expression. It
also simplifies to 320.
1 Point = The student answers one
part correctly but does not attempt
all parts; or the student attempts
to answer all parts of the problem
but does not correctly answer any
part.
0 Points = The student does not