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Exponents and
Roots
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CHAPTER
4
Name
Class
Date
Lesson
Worktext
Remember It?
Copyright © by Holt McDougal. All rights reserved.
Rev. MA.7.A.3.2
Student
Textbook
85
144 –147
4-1 Exponents
MA.8.A.6.1
4-2 Integer Exponents
87 – 94
148 –151
MA.8.A.6.1
4-3 Scientific Notation
95 –102
152 –156
MA.8.A.6.3
4-4 Laws of Exponents
103 –110
157–161
MA.8.A.6.2
4-5 Squares and Square Roots
111 – 118
164 –167
MA.8.A.6.2
4-6 Estimating Square Roots
121 –128
168 –171
MA.8.A.6.4
4-7 Operations with Square Roots
129 –136
172 –175
MA.8.A.6.2
4-8 The Real Numbers
137–144
176 –179
MA.8.G.2.4
4-9 The Pythagorean Theorem
145 –152
180 –183
MA.8.G.2.4
4-10 Applying the Pythagorean
Theorem and Its Converse
153 –160
184 –187
Study It!
163 –165
Write About It!
166
Chapter 4 Exponents and Roots 83
CHAPTER
Benchmark
4
Chapter at a Glance
Vocabulary Connections
LA.8.1.6.5 The student will relate new vocabulary to familiar words.
Key Vocabulary
Vocabulario
Vokabilè
hypotenuse
hipotenusa
ipoteniz
irrational number
número irracional
nonm irasyonèl
perfect square
cuadrado perfecto
kare pafè
Pythagorean Theorem
teorema de Pitágoras
Teorèm Pythagò (Pythagore)
real number
número real
nonm reyèl
scientific notation
notación cientifica
notasyon syantifik
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.
CHAPTER
1. The word irrational contains the prefix ir-, which means “not.” Knowing what you
do about rational numbers, what do you think is true of irrational numbers?
2. The word real means “actual” or “genuine.” How do you think this applies to “real
numbers” that we speak of in math?
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4
84 Chapter 4 Exponents and Roots
4-1
Name
Class
Date
Remember It?
Review skills and prepare for future lesson
lessons.
4-1
Lesson
Exponents (Student Textbook pp. 144–147)
Rev of MA.7.A.3.2
W
Write
it iin exponential form.
4·4·4
43
Identify how many times 4 is used as a factor.
Simplify.
-53
-( 53 )
-( 5 · 5 · 5 ) = -125
Find the product of three 5’s and then make the answer negative.
Write in exponential form.
1. 7 · 7 · 7
2. ( -3 ) · ( -3 )
3. -3 · 3 · 3 · 3
4. __23 · __23 · __23 · __23
5. x · x · x
6. 2n · 2n · 2n
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Simplify.
7. 54
8. ( -1 )9
9. -24
11. (-2)5
10. ( -2 )4
2
12. ( __45 )
13. Evaluate k2 + 3k for k = -2.
Notes
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Chapter 4 Exponents and Roots 85
Notes
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86 Chapter 4 Exponents and Roots
Explore It!
Learn It!
Summarize It!
Name
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Class
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4-2
Date
Explore It!
MA.8.A.6.1 Use exponents…to write
large and small numbers and vice
versa and to solve problems.
Integer Exponents
Explore Patterns in Exponents
A power is a product made of repeated factors. In a
power, an exponent is used to tell how many times a
base is used as a factor.
REMEMBER
3 is the base.
2 is the exponent.
32 = 3 · 3
Activity 1
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1 Complete the table and observe the patterns.
Exponential
Form
Base
Exponent
Number of
Factors
Expanded Form
Value
32
3
2
2
3·3
9
24
2
4
2·2·2·2
6
3
6·6·6
x4
x4
42
2
1
5
a
3
3
125
2
64
Try This
Write the value of the power described.
1. base 2, number of factors 5
2. expanded form 5 · 5 · 5
3. exponent 4, base 3
4. number of factors 4, base 4
4-2 Integer Exponents 87
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Draw Conclusions
5. Is 25 the same as 2 × 5? Explain.
The exponents in Activity 1 were all positive whole numbers. You can use number
patterns to discover how to use negative exponents.
Activity 2
1 Complete the tables. Some of the exponents are negative.
Exponential
Form
Expanded Form
Exponential
Form
Expanded Form
22
2·2
52
5·5
21
2
51
5
20
1
50
1
2-1
1
__
2
5-1
1
__
5
2-2
1
____
2·2
5-2
1
____
5·5
2-3
1
_______
2·2·2
5-3
1
_______
5·5·5
Value
1
5-4
2-5
5-5
1
Try This
Find the value of each power.
6. 2-3
7. 3-2
8. 4-4
Draw Conclusions
10. Explain how to find the value of a number raised to the -5 power.
88 4-2 Integer Exponents
9. 10-4
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2-4
Value
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4-2
Date
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MA.8.A.6.1 Use exponents…to write
large and small numbers and vice
versa and to solve problems.
Integer Exponents (Student Textbook pp. 148–151)
Lesson Objectives
Simplify expressions with negative exponents and evaluate the zero exponent
102
101
100
10-1
10-2
10-3
10 · 10
10
1
1
__
1
______
1
_________
10
10 · 10
10 · 10 · 10
100
10
1
1
__
= 0.1
1
___
= 0.01
1
____
= 0.001
÷ 10
÷ 10
10
÷ 10
100
÷ 10
1000
÷ 10
Example 1
Simplify.
Si
lif W
Write in decimal form.
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A. 10-2
B. 10-1
1
10-2 = ______
10 · 10
1
10-1 = __
10
10-2 =
10-1 =
10-2 =
10-1 =
Check It Out!
Simplify. Write in decimal form.
S
1a. 10-8
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1b. 10-5
4-2 Integer Exponents 89
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Example 2
A. Simplify
A
Si lif 5-3.
1
5-3 = _____
Write the
in the numerator and
in the denominator.
1
5-3 = ___________
5-3 =
Find the product.
B. Simplify ( -10 )-3
1
(-10)-3 = _________
Write the
in the numerator and
in the denominator.
1
(-10)-3 = __________________________
(-10)-3 =
Find the product.
Check It Out!
2a. Simplify ( -3 )-5.
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2b. Simplify ( -4 )-6.
90 4-2 Integer Exponents
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Example 3
Simplify
Si
lif 5 - ( 6 - 4 )-3 + ( -2 )0.
5 - ( 6 - 4 )-3 + ( -2 )0
= 5 - ( 2 )-3 + ( -2 )0
Subtract inside the
=5-(
Evaluate the
)+1
.
.
Add and subtract from left to right.
=
Check It Out!
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3a. Simplify 10 + ( 5 + 3 )-2 + 50.
3b. Simplify ( 7 - 3 )3 + 2-2 - 5.
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4-2 Integer Exponents 91
Explore It!
4-2
Learn It!
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Name
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Date
LA.8.2.2.3 The student will organize
information to show understanding or
relationships…
Integer Exponents
Think and Discuss
1. Express _12 using a negative exponent.
2. Tell whether an integer raised to a negative exponent can ever be greater than 1.
Justify your answer.
3. Get Organized Complete the graphic organizer. Fill in the boxes by writing an
example of an expression with each type of exponent. Show how to simplify each
expression that you write.
Simplifying
Powers
92 4-2 Integer Exponents
Negative
Exponent
Zero
Exponent
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Positive
Exponent
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4-2
Date
MA.8.A.6.1 Use exponents...to write
large and small numbers and vice
versa and to solve problems.
Integer Exponents
Simplify. Write in decimal form.
1. 10-7
2. 10-12
Simplify.
3. ( -4 )3
4. ( -6 )-2
5. 9-2
7. ( -8 )-4
8. ( -7 )4
9. -65
-5
14. 78 - 6-2
-4
15. ( 5 - 1 )2 - 3
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17. 6-2 + 10 · 2
10. 3-8
12. ( -4 )-4 + 7
11. 33 - 15
13. 72 - 2
6. -142
+3
4
16. ( 6-3 )-3 · 4
2
18. -3(4-3 + 92)
19. -43 + 2-5
20. ( -2 + 8 )-3 + 24
21. ( 1-3 )5 · ( -4 )-2
22. 32 + 1-2 · 2-3
Evaluate each expression for the given value of the variable.
23. x -3 for x = -3
24. 6y-2 for y = -4
25. -w 4 + 17 for w = 3
26. -4( s-2 ) for s = -1
27. ( -3t ) 2 for t = 4
28. -b-2 - 6 for b = 4
29. Write an expression for the product of eight and x, raised to the negative third
power. Then evaluate the expression for x = –4.
30. Write an expression for the difference of four and x, raised to the negative second
power. Then evaluate the expression for x = 2.
4-2 Integer Exponents 93
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4-2
Learn It!
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Date
MA.8.A.6.1 Use exponents…to write
large and small numbers and vice
versa and to solve problems.
Integer Exponents
1. The weight of one dust particle is 10-7
gram. Write this measure in standard
notation.
2. The northern yellow bat is one of Florida’s
larger bat species. An adult has a wingspan
of about 14 inches and weighs between
3( 2 )-3 and 3( 2 )-2 ounces. Simplify these
expressions.
3. Recall that the formula for the area of
a circle is A = πr 2. How can you use
negative exponents to solve this equation
for π ?
5. A ruby-throated hummingbird breathes
2 × 53 times per minute while at rest. Write
the simplest form for this number of
breaths per minute.
94 4-2 Integer Exponents
Unit
Size in meters
centimeter (cm)
10-2 m
millimeter (mm)
10-3 m
micrometer (μm)
10-6 m
nanometer (nm)
10-9 m
Angstrom (Å)
10-10 m
6. Human eyes can see a resolution of about
100 μm. Write this measure in meters.
7. The size of a bacterium is about 50 nm.
Write this measure in meters.
8. Gridded Response
An Angstrom is
equal to 100,000
femtometers. What
exponent of base 10
is used to express the
size of a femtometer in
meters?
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4. A ruby-throated hummingbird weighs
about 3-2ounce. Simplify 3-2.
Use the table for 6–8.
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Class
4-3
Date
MA.8.A.6.1 Use exponents and
scientific notation to write large and
small numbers and vice versa and to
solve problems.
Explore It!
Scientific Notation
Explore Products and Powers of Ten
You will investigate how to use powers of 10 to help you write very large numbers.
Activity 1
1 Complete the table by writing the value of each power of 10.
100
Power of 10
Value
101
102
1
103
104
100
2 Complete the table by finding the indicated product or factor.
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Factors
Factor × Power of Ten
256 × 10
256 × 10
256 × 100
256 × 10
256 × 1000
256 × 10
25.6 ×
25.6 ×
25.6 × 100
25.6 ×
× 1000
2.56 ×
× 100
2.56 ×
× 1000
6.75 ×
Product
256
×
25,600
×
25.6
2.56 ×
×
2560
×
3290
6.75 ×
6,750,000
Try This
Find the indicated power of 10.
1. 3.49 ×
3. 8.654 ×
= 349
= 865.4
2. 2.19 ×
= 21,900
4. 7.094 ×
= 7,094
4-3 Scientific Notation 95
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You can also use powers of 10 to help you write very small numbers.
Activity 2
1 Complete the table by writing the decimal value of each power of 10.
100
Power of 10
Value
10-1
10-2
1
10-3
10-4
0.01
2 Complete the table by finding the indicated product or factor.
Factors
Factor × Power of Ten
256 × 0.1
256 × 10
256 × 0.01
256 × 10
256 × 0.001
256 × 10
25.6 ×
25.6 ×
25.6 × 0.01
25.6 ×
× 0.001
2.56 ×
× 0.01
Product
2.56
×
0.0256
×
0.256
2.56 ×
×
× 0.001
6.75 ×
0.00000675
Try This
Find the indicated power of 10.
5. 3.49 ×
= 0.0349
6. 8.654 ×
= 0.08654
7. 2.19 ×
= 0.000219
8. 7.094 ×
= 0.007094
Draw Conclusions
9. Describe how to multiply by a power of 10 with a positive exponent.
10. Describe how to multiply by a power of 10 with a negative exponent.
96 4-3 Scientific Notation
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6.75 ×
0.00329
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Class
4-3
Date
MA.8.A.6.1 Use exponents and
scientific notation to write large and
small numbers and vice versa and to
solve problems.
Learn It!
Scientific Notation (Student Textbook pp. 152–156)
Lesson Objectives
Express large and small numbers in scientific notation and compare two
numbers written in scientific notation
Vocabulary
scientific notation
Example 1
Write
W
it each
h number in standard notation.
A. 1.35 × 105
1.35 × 105
105 =
1.35 ×
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Think: Move the decimal right
places.
B. 2.7 × 10-3
2.7 × 10-3
10-3 =
2.7 ×
2.7
1000
by the reciprocal.
Think: Move the decimal
3 places.
C. -2.01 × 104
-2.01 × 104
-2.01 ×
104 =
Think: Move the decimal right
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places.
4-3 Scientific Notation 97
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Write each number in standard notation.
1b. 1.9 × 10-8
1a. 6.95 × 106
1c. -3.8 × 10-5
Example 2
W
it 0
0070 in scientific notation.
Write
0.00709
0.00709
Move the decimal to get a number between
7.09 × 10
Set up
and
.
notation.
Think: The decimal needs to move left to change 7.09 to
0.00709, so the exponent will be
So 0.00709 written in scientific notation is
Check
×
×
places.
.
= 7.09 × 0.001 = 0.00709
Check It Out!
2. Write 0.000811 in scientific notation.
98 4-3 Scientific Notation
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Think: The decimal needs to move
.
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Example 3
A pencil
il is
i 18.7
18 cm long. If you were to lay 10,000 pencils of this length
end-to-end, how many millimeters long would they be? Write the answer in
scientific notation.
1 cm =
mm, so 18.7 cm =
187 mm × 10,000
mm.
Find the total length.
.
Think: The decimal needs to move
In scientific notation, the pencils end-to-end would be
places to the
.
mm long.
Check It Out!
3. An oil rig can hoist 2,400,000 pounds with its main derrick. It distributes the
3
weight evenly between 8 wire cables. What is the weight that each wire cable
can hold? Write the answer in scientific notation.
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Example 4
O cell
One
ll h
has a diameter of approximately 4.11 × 10-5 meters. Another cell
has a diameter of 1.5 × 10-5 meters. Which cell has a greater diameter?
4.11 × 10-5
?
1.5 × 10-5
10-5
10-5
Compare powers of
4.11
1.5
Since powers of 10 are equal, compare the values between 1 and 10.
4.11 × 10-5
The
.
1.5 × 10-5
has a greater diameter.
Check It Out!
4. A certain cell has a diameter of approximately 5 × 10-3 meters. A second cell
has a diameter of 5.11 × 10-3 meters. Which cell has a greater diameter?
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4-3 Scientific Notation 99
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LA.8.2.2.3 The student will organize
information to show understanding or
relationships…
Scientific Notation
Think and Discuss
1. Explain the benefits of writing numbers in scientific notation.
2. Describe how to write 2.977 × 106 in standard notation.
Writing Numbers in Scientific Notation
Positive Number
Greater than 0, less than 1
Greater than or equal to 1, less than 10
Greater than or equal to 10
100 4-3 Scientific Notation
Exponent
Example
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3. Get Organized Complete the graphic organizer. Tell whether the exponent
is positive, negative, or zero when each type of positive number is written in
scientific notation.
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Date
MA.8.A.6.1 Use exponents and
scientific notation to write large and
small numbers and vice versa and to
solve problems.
Scientific Notation
Write each number in standard notation.
1. 6.12 × 102
2. 7.9 × 10-3
3. 4.87 × 104
4. 9.3 × 10-2
5. 8.06 × 103
6. 5.7 × 10-4
7. 3.17 × 10-5
8. 9.00613 × 10-2
9. 9.85 × 10-5
10. 6.004 × 107
12. 1.48 × 10-6
11. 8.23 × 104
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Write each number in scientific notation.
13. 108,000,000
14. 0.5943
15. 42
16. 0.0000673
17. 0.0056
18. 6004
19. 0.00852
20. 24,631,500
21. 89,450
22. 0.005702
23. 8,005,000,000
24. 0.00012805
Compare. Write >, <, or =.
25. 1.7 × 10-5
1.6 × 10-4
26. 5.8 × 106
9.01 × 104
27. -4 × 108
-3 × 108
28. The mass of Earth is approximately 5,980,000,000,000,000,000,000,000 kilograms.
Write this number in scientific notation.
29. The mass of a specific dust particle is 7.53 × 10-10 grams. Write this number in
standard notation.
4-3 Scientific Notation 101
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Date
MA.8.A.6.1 Use exponents and
scientific notation to write large and
small numbers and vice versa and to
solve problems.
Apply It!
Scientific Notation
1. In June 2001, the Intel Corporation
announced that they could produce a
silicon transistor that could switch on and
off 1,500,000,000,000 times per second.
Express the speed of the transistor in
scientific notation.
One light-year is approximately equal
to 5,870,000,000,000 miles. Use this
information and the table for 8–11. Write
your answers in scientific notation.
Distance From Earth To Stars
Star
Sirius
2. With this transistor, computers will be able
to do 1 × 109 calculations in the time it
takes to blink your eye. Express the number
of calculations in standard notation.
Canopus
Alpha Centauri
Vega
Constellation
Canis Major
Carina
Distance
(light-years)
8
650
Centaurus
4
Lyra
23
8. How far in miles is Sirius from Earth?
3. The elements in this fast transistor are
20 nanometers long. A nanometer
1
is __________
of a meter. Express the length
1,000,000,000
of an element in the transistor in meters in
scientific notation.
4. micro; 8 microseconds
5. nano; 5 nanoseconds
6. pico; 6 picoseconds
7. femto; 2 femtoseconds
102 4-3 Scientific Notation
10. How much closer is Alpha Centauri from
Earth than Sirius?
11. Short Response Explain how to use
scientific notation to express a light-year
in miles.
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Use a dictionary to find the meanings of
each numerical prefix. Then write the given
measure in seconds using scientific notation.
9. How much farther is Canopus from Earth
than Sirius?
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4-4
Date
MA.8.A.6.3 Simplify real number
expressions using the laws of
exponents.
Laws of Exponents
Multiply and Divide Powers
You can use patterns in tables of numbers to discover rules for
multiplying and dividing numbers written in exponential form.
Activity 1
Complete the table. Use the information in Column 2 to write the product as
a power in Column 3.
Product
Factors of Product
Exponential Form
Sum of Exponents
in Column 1
43 · 42
(4 · 4 · 4) · (4 · 4)
45
3+2=5
24 · 25
(2 · 2 · 2 · 2) · (2 · 2 · 2 · 2 · 2)
29
4+5=9
32 · 33
(3 · 3) · (3 · 3 · 3)
35
63 · 63
(6 · 6 · 6) · (6 · 6 · 6)
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105 · 102
26 · 21
53 · 54
71 · 78
Try This
Write each product as a single power. Example: 32 · 33 = 35
1. 23 · 24
2. 62 · 62
3. 35 · 31
4. 82 · 85
Draw Conclusions
5. Compare: Look at the exponent in the third column and the sum in the fourth
column. How are they alike?
6. Describe how you can multiply powers with the same base, like those in the
first column.
4-4 Laws of Exponents 103
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Activity 2
Complete the table. Be sure to eliminate common factors in Column 2.
Quotient
Factors of Product
Quotient
Written as a
Power
Difference of
Exponents in
Column 1
26
__
22
2·2·2·2·2·2
______________
2·2
24
6-2=4
35
__
31
3·3·3·3·3
____________
3
34
5-1=4
74
__
72
7·7·7·7
_________
7·7
58
__
55
47
__
46
24
__
21
64
__
60
96
__
91
Try This
7
3
2
Write the quotient as a single power. Example: __
5 =3
3
86
8. __
2
8
108
10. ___
103
Draw Conclusions
11. Compare: Look at the exponent in the third column and the difference in the
fourth column. How are they alike?
12. Describe how you can divide powers with the same base.
104 4-4 Laws of Exponents
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54
7. __
53
95
9. __
93
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4-4
Date
Learn It!
MA.8.A.6.3 Simplify real number
expressions using the laws of
exponents.
Laws of Exponents (Student Textbook pp. 157–161)
Lesson Objectives
Apply the laws of exponents
Example 1
M lti l W
Multiply.
Write the product as one power.
A. 66 · 63
66 + 3
C.
25 · 2
25 + 1
Check It Out!
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exponents.
B. n5 · n7
n5 + 7
exponents.
D. 244 · 244
exponents.
244 + 4
exponents.
M
Multiply.
Write the product as one power.
1a. 42 · 44
1b. x4 · x2
1c. 15 · 152
1d. p2 · p2
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4-4 Laws of Exponents 105
Learn It!
Explore It!
Summarize It!
Practice It!
Apply It!
Example 2
Divide.
Di
id W
Write
it the quotient as one power.
75
A. __
3
7
75 - 3
exponents.
x10
B. ___
9
x
x10 - 9
Subtract
.
Think: x1 =
Check It Out!
Divide. Write the quotient as one power.
n8
2b. __
5
99
2a. __
2
n
9
Si lif
Simplify.
2
A. ( 54 )
9
B. ( 67 )
54 · 2
C.
exponents.
-3
12 · -3
106 4-4 Laws of Exponents
exponents.
-20
D. ( 172 )
( ( _23 )12 )
( _23 )
67 · 9
exponents.
172 · -20
exponents.
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Example 3
Explore It!
Learn It!
Check It Out!
-1
3a. ( 72 )
Summarize It!
Practice It!
Apply It!
Simplify.
2
3b. ( 7-1 )
3c.
4 -2
-3
3d. ( 5-2 )
[ (__15 ) ]
Example 4
Th
The speed
d off sound at sea level is 3.4029 × 102 meters per second. A ship that
is 5 kilometers offshore sounds its horn. About how many seconds will pass
before a person standing on shore will hear the sound? Write your answer in
scientific notation.
distance = rate × time
5 km = ( 3.4029 × 102 ) × t
_______ = ( 3.4029 × 102 ) × t
__________ = ( 3.4029 × 102 ) × t
Copyright © by Holt McDougal. All rights reserved.
3.4029 × 102 × t
5 × 103
_________________
= __________________
Write 5 km as meters.
Write
in scientific notation.
Divide both sides by
×
=t
Write as a product of quotients.
1.469 ×
≈t
Simplify each quotient.
.
It would take about 1.5 × 101 seconds for the sound to reach the shore.
Check It Out!
4. The diameter of a red blood cell is about 7.6 × 10-4 millimeters. Thai has a slide
with a 2-cm drop of red blood cells on it. Approximately how many cells are on
the slide? Write your answer in scientific notation. (Hint: Find the ratio of the
size of the drop, in millimeters, to the size of one cell.)
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4-4 Laws of Exponents 107
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4-4
Learn It!
Summarize It!
Name
Practice It!
Class
Summarize It!
Apply It!
Date
LA.8.2.2.3 The student will organize
information to show understanding or
relationships …
Laws of Exponents
Think and Discuss
1. Explain why the exponents cannot be added in the product 143 · 183.
2. List two ways to express 45 as a product of powers.
3. Get Organized Complete the graphic organizer. Fill in the boxes by writing an
example that illustrates each Law of Exponents.
Laws of
Exponents
108 4-4 Laws of Exponents
Dividing Powers
with the Same Base
Raising a Power
to a Power
Copyright © by Holt McDougal. All rights reserved.
Multiplying Powers
with the Same Base
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Class
Practice It!
4-4
Date
MA.8.A.6.3 Simplify real number
expressions using...the laws of
exponents.
Laws of Exponents
Multiply. Write the product as one power.
1. 106 × 109
2. a8 × a6
3. 156 × 1512
4. 1112 × 117
5. (-w)8 × (-w)12
6. (-12)18 × (-12)13
7. 1310 × 1315
8. w14 × w12
Divide. Write the quotient as one power.
( -13 )14
10. _______
9
a25
9. ___
18
( -13 )
1811
____
12. 5
18
22
21
14. ____
2120
258
16. ___
253
a
12
____
11. 14 8
14
15
19
13. ____
194
17
(-x)
15. _____7
(-x)
Copyright © by Holt McDougal. All rights reserved.
Write the product or quotient as one power.
17. r9 × r8
1620
18. ____
10
x15
19. ___
9
20. ( -17 )8 × ( -17 )7
m16
21. ___
10
22. ( -b )21 × ( -b )14
16
m
x
23. Hampton has a baseball card collection of 56 cards. He organizes
the cards into boxes that hold 54 cards each. How many boxes will
Hampton need to hold the cards? Write the answer as one power.
24. Write the expression for a number used as a factor seventeen times
being multiplied by the same number used as a factor fourteen
times. Then write the product as one power.
3
25. After 3 hours, a bacteria colony has ( 252 ) bacteria present. How many
bacteria are in the colony? Write your answer in standard form.
4-4 Laws of Exponents 109
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4-4
Learn It!
Name
Summarize It!
Class
Apply It!
Apply It!
Practice It!
Date
MA.8.A.6.3 Simplify real number
expressions using the laws of
exponents.
Laws of Exponents
1. A researcher separated her fruit flies into
22 jars. She estimates that there are 210 fruit
flies in each jar. How many fruit flies does
the researcher have in all?
2. Suppose a researcher tests a new method of
pasteurization on a strain of bacteria in his
laboratory. If the bacteria are killed at a rate
of 89 per second, how many bacteria would
be killed after 82 second?
3. A satellite orbits the earth at about 134
km per hour. How long would it take to
complete 24 orbits, which is a distance of
about 135 km? (Hint: Use d = rt, distance
equals rate times time.)
Use the table for 6 and 7. The table describes
the number of people involved at each
level of a pyramid pattern. In this pyramid
pattern, each individual recruits 5 others to
participate, who in turn recruit 5 others, and
so on.
Pyramid Pattern
Level
Total Number of People
1
5
2
52
3
53
4
54
6. How many levels will it take to exceed
100,000 people?
4. The side of a cube is 34 centimeters long.
What is the volume of the cube?
(Hint: V = s3)
5. The wavelengths of electromagnetic
radiation vary greatly. Green light has a
wavelength of about 5.1 × 10-7 meters.
The wavelength of a U-band radio wave is
2.0 × 10-2 meters. About how many times
greater is the wavelength of a U-band radio
wave than that of a green light? Justify your
answer.
110 4-4 Laws of Exponents
8. Short Response 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? Write your answer in
scientific notation. Show your work.
Copyright © by Holt McDougal. All rights reserved.
7. How many times more people will be
involved at level 6 than at level 2?
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Name
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Class
4-5
Date
MA.8.A.6.2 Make reasonable
approximations of square roots…,
and use them to estimate solutions
to problems…
Explore It!
Squares and Square Roots
Relate Squares and Square Roots
To square a number means to multiply the number by
itself.
side = 5
5 squared = 5 × 5 = 52 = 25
25 is the area of a square that is 5 units long on a side.
In these activities, you will investigate the relationship
between squares and square roots.
area = 52 = 25
Activity 1
1 Use square tiles. Make a square that measures 6 tiles on a side.
How many tiles did you use?
2 Make a square that measures 4 tiles on a side. How many tiles did
you use?
Copyright © by Holt McDougal. All rights reserved.
3 On the grid at the right, draw squares 3 units on a side and 7 units
on a side. Inside each square, write the area of the square.
4 Complete the table.
Number
1
2
Number Squared
12
22
3
4
5
6
7
8
9
10
Expanded Form
Evaluate
Try This
Evaluate each square.
1. 122
2. 152
3. 13 · 13
4. 20 squared
Draw Conclusions
5. Explain why 42 equals 16 and does not equal 8.
4-5 Squares and Square Roots 111
Explore It!
Learn It!
Summarize It!
Practice It!
Apply It!
When you find two equal factors of a number, you have found a square root
of the number.
7 × 7 = 49, so 7 is a square root of 49. Since (-7) · (-7) = 49, -7 is also a
square root of 49.
Use a radical symbol √to indicate the nonnegative square root: √
49 = 7.
Activity 2
1 Use 16 square tiles to make a square.
How many tiles are on each side of the square?
2 What happens if you try to make a square using 20 square tiles?
3 Complete the table.
Number
1
4
Number Squared
1
16
Square Root of Number
1
2
9
16
25
64
625
6
10
number to indicate the negative square root of a number.
Use the symbol -√
4 Complete the table.
1
4
√
Number
1
2
-1
-2
-√
Number
9
25
49
144
10
-8
Try This
Evaluate each square root.
6. √
16
7. -√
36
8. √
81
Draw Conclusions
10. Explain why both 10 and -10 are square roots of 100.
112 4-5 Squares and Square Roots
9. -√
121
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Number
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4-5
Date
MA.8.A.6.2 Make reasonable
approximations of square roots…,
and use them to estimate solutions
to problems…
Learn It!
Squares and Square Roots (Student Textbook pp. 164–167)
Lesson Objective
Find square roots
Vocabulary
square root
principal square root
perfect square
Example 1
Copyright © by Holt McDougal. All rights reserved.
Find
Fi
d th
the ttwo square roots of each number.
A. 49
√
49 =
- √
49 =
is a square root, since 7 · 7 =
.
is also a square root, since (-7) · (-7) =
.
B. 100
√
100 =
- √
100 =
is a square root, since 10 · 10 =
.
is also a square root, since (-10) · (-10) =
.
C. 225
√
225 =
- √
225 =
is a square root, since 15 · 15 =
.
is also a square root, since (-15) · (-15) =
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.
4-5 Squares and Square Roots 113
Learn It!
Explore It!
Check It Out!
Summarize It!
Practice It!
Apply It!
Find
F
the two square roots of each number.
1a. 81
1b. 144
1c. 324
1d. 361
Example 2
A square window
i
has an area of 169 square inches. How wide is the window?
Find the square root of
square root; a negative length has no meaning.
= 169
So √
169 =
The window is
inches wide.
Check It Out!
2. A square window has an area of 225 square inches. How wide is the window?
114 4-5 Squares and Square Roots
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Copyright © by Holt McDougal. All rights reserved.
Use the
to find the length of the window.
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Summarize It!
Practice It!
Apply It!
Example 3
Simplify
Si
lif each
h expression.
A. 3 √
36 + 7
3 √
36 + 7 = 3(
)+7
=
+7
=
B.
root.
Multiply.
Add.
25 + __
3
__
√
4
16
√
25
25 + __
3 = _____
__
+ __34
√
4
16
√
16
=
=
Check It Out!
Copyright © by Holt McDougal. All rights reserved.
Evaluate the
25 as
__
Rewrite √
16
+ __34
Evaluate the
.
roots.
Add.
S
Simplify
each expression.
3a. 2 √
121 + 9
3b.
16 + __
2
__
√
36
3
3c. -5 √
336 + 25
3d.
√
12 · 2 √
48
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4-5 Squares and Square Roots 115
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4-5
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Name
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Class
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Date
Summarize It!
LA.8.2.2.3 The student will organize
information to show understanding or
relationships…
Squares and Square Roots
Think and Discuss
1. Describe what is meant by a perfect square. Give an example.
2. Explain how many square roots a positive number can have. How are these
square roots different?
3. Decide how many square roots 0 has. Tell what you know about square roots of
negative numbers.
Copyright © by Holt McDougal. All rights reserved.
4. Get Organized Complete the graphic organizer. Fill in the boxes by writing the
principal square root and negative square root of 36. For each of these, write the
square root using a radical symbol ( √ ) and as an integer.
Square Roots
of 36
Negative
Square Root
Principal
Square Root
Radical
Integer
116 4-5 Squares and Square Roots
Radical
Integer
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Name
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4-5
Date
MA.8.A.6.2 Make reasonable
approximations of square roots...,
and use them to estimate solutions to
problems...
Practice It!
Squares and Square Roots
Find the two square roots for each number.
1. 16
2. 9
3. 64
4. 121
5. 36
6. 100
7. 225
8. 400
Copyright © by Holt McDougal. All rights reserved.
Evaluate each expression.
9. √
27 + 37
10. √
41 + 59
11.
13. 3 √
81 + 19
14. 25 - √
25
15. √
169 - √
36
16. √
196 + 25
√
81
17. ____
9
18. -4.9 √
64
√
225
19. _____
√
100
20. _____
-2.5
√
122 - 41
√
25
12. √
167 - 23
21. Find the product of six and the sum of the square roots of 100 and 225.
22. Explain how you can verify that √
289 is 17 without using a calculator.
23. If a replica of the ancient pyramids was built with a base area
of 1024 in2, what would be the length of each side? (Hint: s = √
A)
24. The maximum displacement speed of a boat is found using the
formula: Max Speed (km/h) = 4.5 √
waterline length (m) .
Find the maximum displacement speed of a boat that has a
waterline length of 9 meters.
4-5 Squares and Square Roots 117
Explore It!
4-5
Learn It!
Name
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Practice It!
Class
Apply It!
Date
MA.8.A.6.2 Make reasonable
approximations of square roots…,
and use them to estimate solutions
to problems…
Apply It!
Squares and Square Roots
Use the table for 1–3.
A carpenter wants to use as many of her
82 small wood squares as possible to make
a large square inlaid box lid. Use this
information for 6 and 7.
Amateur Wrestling
Square Mat Sizes
Division
Size
Home Use - small
100 ft2
Home Use - large
144 ft2
High School Competition
1444 ft2
NCAA College Competition
1764 ft2
1. What is the length of each side of
the wrestling mat for NCAA College
competition?
2. What is the length of each side of the
wrestling mat for High School Competition?
4. A middle school plans to purchase a 32-ft
by 32-ft practice mat. If the estimated cost is
$3.50–$4.50 per square foot, how much can
they expect to pay for the new mat?
5. The Japanese art of origami requires folding
square pieces of paper. Elena begins with a
large sheet of square paper that is 169 in2.
How many 4-in. × 4-in. squares can she cut
out of the paper?
118 4-5 Squares and Square Roots
7. How many more small wood squares would
the carpenter need to make the next larger
possible square box lid?
8. When the James family moved into a new
house they had a square area rug that was
132 square feet. In their new house, there
are three bedrooms. Bedroom one is 11 feet
by 11 feet. Bedroom two is 10 feet by 12 feet,
and bedroom three is 13 feet by 13 feet. In
which bedroom will the rug fit?
9. Gridded Response
A box of tiles contains
12 tiles. If you tile a
square area using
whole tiles, how many
tiles will you have left
from the box?
Copyright © by Holt McDougal. All rights reserved.
3. Compare the mats for home use. How
much smaller is one side of a small mat
than one side of a large mat?
6. How many squares can the carpenter use?
How many squares would she have left?
4-2
Name
Class
THROUGH
Date
4-5
Got It?
Ready to Go On?
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Quiz for Lessons 4-2 through 4-5
4-2
Integer Exponents (Student Textbook pp. 148–151)
Simplify.
1. 10-6
2. ( -3 )-4
3. -6-2
4. 5-1 + 3( 5 )-2
5. -4-3 + 20
6. 3-2 - ( 60 - 6-2 )
4-3
Scientific Notation (Student Textbook pp. 152–156)
Write each number in scientific notation.
7. 0.00000015
8. 99,980,000
9. 0.434
10. 100
Write each number in standard notation.
Copyright © by Holt McDougal. All rights reserved.
11. 1.38 × 105
12. 4 × 106
13. 1.2 × 10-3
14. 9.37 × 10-5
15. According to the US Census Bureau, the population of Florida in
2000 was nearly 16 million people, and the per capita income was
approximately $21,500. Write the estimated total income in 2000 for
Florida residents in scientific notation.
4-4
Laws of Exponents (Student Textbook pp. 157–161)
Simplify. Write the product or quotient as one power.
16. 93 · 95
510
17. ___
10
18. q9 · q6
19. 33 · 3-2
-2
20. ( 33 )
0
21. ( 42 )
4
22. ( -x2 )
5
23. ( 4-2 )
5
24. The mass of the known universe is about 1023 solar masses, which
is 1050 metric tons. How many metric tons is one solar mass?
4-5
Squares and Square Roots (Student Textbook pp. 164–167)
Find the two square roots of each number.
25. 16
26. 9801
27. 10,000
28. 529
29. If Jan’s living room is 20 ft × 16 ft, will a square rug with an area of 289 ft2 fit?
Justify your answer.
Chapter 4 Exponents and Roots 119
4-2
THROUGH
4-5
Name
Class
Date
Connect It!
MA.8.A.6.1, MA.8.A.6.3,
MA.8.A.6.4
Connect the concepts of Lessons 4-2 through 4-5
Match Up
Work with a partner to play this matching game.
2-4
1. Write each of the following expressions on an
index card.
2. Shuffle the cards. Lay them out face down on
the desk in front of you.
42 · 40
50
1
16
169
168
410
48
1 2
16
2.5 × 10-1
15
(22)2
(32)2
4. The game ends when there are no cards left.
The winner is the player who collects the most
cards.
Find a path that lands on each disk exactly
once. You may start at any disk and then hop
to any disk next to it. When you hop, you
must always move to a greater number.
1. Draw your path along the disks.
Think About The Puzzler
2. Describe any strategies you used to find the path.
120 Chapter 4 Exponents and Roots
-6
8
12 · 12
1.6 × 101
1.44 ×
1
16
(22)3
1.6 ×
23 · 22
210
27
1.5 × 106
104
2-3
7.3 × 10-8
(-2)2
10 2
2-2
14
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Hop To It!
24
37 · 3-3
3. You and your partner should take turns flipping
over two cards. If the expressions on the cards
are equal, the player who made the match keeps
the cards and takes another turn. If the cards
are not a match, the player turns the cards facedown and the other player takes a turn.
5. Describe any strategies you used to help you
play the game.
22
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Name
Practice It!
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Class
4-6
Date
MA.8.A.6.2 Make reasonable
approximations of square roots…,
and use them to estimate solutions to
problems…
Explore It!
Estimating Square Roots
Make Square Root Approximations
A perfect square is a number that has integers as its
square roots. 25 is a perfect square because √
25 = 5 and
√
- 25 = -5. The square roots, 5 and -5, are integers.
You can use perfect squares to estimate square roots of
numbers that are not perfect squares.
12 = 1
72 = 1
22 = 4
82 = 4
32 = 9
92 = 9
42 = 16
102 = 16
52 = 25
112 = 25
2
6 = 36
122 = 36
Activity 1
1 Estimate the location of √
11 on the number line.
Think: √
9 = 3 and √
16 = 4. So, √
11 must lie between 3 and 4.
2
9
16
3
4
5
Copyright © by Holt McDougal. All rights reserved.
2 Estimate the locations of √
5 , √
20 , √
41 , √
68 , and √
90 on the number line. Write
the square root along with an arrow pointing to its approximate location.
3.02 = 9
1
2
3
4
5
6
7
8
9
10
3 In Step 1, you estimated √
11 as a number between 3 and 4. You can make a
closer estimate by squaring numbers between 3 and 4. Use the table at the
right. Between what two consecutive numbers, written to tenths,
does √
11 lie?
3.12 = 9.61
3.22 = 10.24
3.32 = 10.89
3.42 = 11.56
3.52 = 12.25
and
4 Use your calculator to find each square.
6.02 =
6.12 =
6.22 =
6.32 =
6.42 =
6.52 =
6.62 =
6.72 =
6.82 =
6.92 =
5 Use your answers to Step 4 to estimate: Between what two consecutive
numbers, written to tenths, does √
45 lie?
and
4-6 Estimating Square Roots 121
Explore It!
Learn It!
Summarize It!
Practice It!
Apply It!
Try This
Between what two consecutive whole numbers does the square root lie?
1. √
70
2. √
35
3. √
6
4. √
52
Draw Conclusions
5. In Step 3 of Activity 1, you found two numbers, written to tenths, between
which √
11 lies. How could you find two numbers, written to hundredths,
between which √
11 lies?
In Activity 2, you can use your calculator to estimate square roots with great accuracy.
Activity 2
1 Complete the table using a calculator. Round square roots to the nearest
thousandth.
Number
Square Root
9
10
11
12
13
14
3.000
15
16
4.000
Solve using a calculator. Round square roots to the nearest thousandth.
3 the length of a side of a square photograph with area 39 in2
4 the length of a side of a square postage stamp with area 172 mm2
Try This
Find the square root to the nearest thousandth.
6. √
125
7. √
279
8. √
432
9. √
300
Draw Conclusions
10. How could you find a number that has a square root that is an integer?
Give an example.
122 4-6 Estimating Square Roots
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2 the length of a side of a square with area 76 cm2
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4-6
Date
MA.8.A.6.2 Make reasonable
approximations of square roots…,
and use them to estimate solutions to
problems…
Learn It!
Estimating Square Roots (Student Textbook pp. 168–171)
Lesson Objective
Estimate square roots and solve problems using square roots
Example 1
E h square root is between two consecutive integers. Name the integers.
Each
Explain your answer.
A.
√
55
Think: Which
72 =
49 < 55
82 =
64 > 55
√
55 is between
and
Copyright © by Holt McDougal. All rights reserved.
B. -√
90
because
is between
Think: Which perfect
(-9)2 =
81 < 90
(-10)2 =
100 > 90
90 is between
-√
Check It Out!
squares are closest to 55?
and
and
.
are closest to 90?
because
is between
and
.
Each square root is between two consecutive integers. Name
E
tthe integers. Explain your answer.
1a. √
80
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1b. √
105
4-6 Estimating Square Roots 123
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Practice It!
Apply It!
Example 2
Y wantt tto sew a fringe on a square tablecloth with an area of 500 square
You
inches. Calculate the length of each side of the tablecloth and the length of fringe
you will need to the nearest inch.
List perfect squares near 500.
,
441,
, 576.
Find the perfect squares nearest 500.
< 500 <
Find the square roots of the perfect squares.
√
<
√
500 <
<
√
500 <
500 is closer to
√
than 529, so
√
500 is closer to
.
√
500 Each side of the tablecloth is about
inches.
Now estimate the length around the tablecloth.
·4⫽
inches of fringe.
Check It Out!
2. You want to build a fence around a square garden that is 250 square feet.
Calculate the length of one side of the garden and the total length of the fence, to
the nearest foot.
124 4-6 Estimating Square Roots
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You will need about
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Example 3
√
Estimate
E
i
141 to the nearest hundredth.
14
Step 1: Find the value of the whole number.
< √
141 <
Find the perfect squares nearest 141, and order the square
roots of the perfect squares.
< √
141 <
The number will be between
The whole number part of the answer is
and
.
.
Step 2: Find the value of the decimal.
141 - 121 =
Find the difference between the given number, 141, and
the lower perfect square.
144 - 121 =
Find the difference between the greater perfect square
and the lower perfect square.
≈
Write the difference as a ratio, and find the approximate
decimal value.
Step 3: Find the approximate value.
+
=
Combine the whole number and decimal.
Copyright © by Holt McDougal. All rights reserved.
The approximate value of √
141 to the nearest hundredth is
.
Check It Out!
3. Estimate √
3
154 to the nearest hundredth.
Example 4
Use a calculator
U
l l
to find √
600 . Round to the nearest tenth.
Using a calculator, √
600 ≈
...
Rounded, √
600 is
.
Check It Out!
4. Use a calculator to find √
800 . Round to the nearest tenth.
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4-6 Estimating Square Roots 125
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Date
LA.8.2.2.3 The student will organize
information to show understanding or
relationships…
Estimating Square Roots
Think and Discuss
1. Discuss whether 9.5 is a good first guess for √
75 .
2. Determine which square root or roots would have 7.5 as a good first guess.
3. Get Organized Complete the graphic organizer. Describe each method and
give an example of each.
Estimating Square Roots
To the nearest tenth
126 4-6 Estimating Square Roots
With a calculator
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Between two integers
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4-6
Date
MA.8.A.6.2 Make reasonable
approximations of square roots…,
and use them to estimate solutions
to problems…
Practice It!
Estimating Square Roots
Each square root is between two consecutive integers. Name the integers.
Explain your answer.
1. √
51
2. √
39
3. √
240
4. √
155
Estimate each square root to the nearest hundredth.
5. √
33
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9. √
51
6. √
15
10. √
8
7. √
37
8.
√
24
11. √
148
12. √
102
2
15. ( √
15 )
16. √
152
Simplify each expression.
13. √
92
2
14. ( √
9)
17. Squaring and taking the square root are said to be inversely related.
Explain what this means.
18. The area of a square tetherball court is 260 ft2. What is the
approximate length of each side of the court? Find your answer to
the nearest foot.
19. Brian jogs one time around a square park with an area of 5 mi2.
About how far does Brian jog?
20. Steve wants to make a curtain to cover a square window with an
area of 12 ft2. About how long should each side of the curtain be?
4-6 Estimating Square Roots 127
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Class
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Date
MA.8.A.6.2 Make reasonable
approximations of square roots…,
and use them to estimate solutions
to problems...
Apply It!
Estimating Square Roots
The distance d in kilometers to the horizon
can be found using the formula
d = 112.88 √
h , where h is the height in
kilometers above the ground. Use this
information for 1–4. Estimate each distance
and show your work. Then use a calculator
to find an approximation to the nearest
kilometer.
1. How far is it to the horizon when you are
standing on the top of Mt. Everest, a height
of 8.85 km?
You can find the approximate speed of a
vehicle that leaves skid marks before it stops.
The table shows the minimum speed, in
mi/h, that a vehicle was traveling before the
brakes were applied. Use this table for 5–8.
Length of Skid
Marks L (ft)
Minimum Speed S (mi/h)
10
0.7( 10 ) ≈ 15
5.5√
20
5.5√
0.7( 20 ) ≈ 21
30
5.5√
0.7( 30 ) ≈ 25
Given the length of a vehicle’s skid mark
before stopping, find the minimum speed of
the vehicle before it stopped. Round to the
nearest mile per hour.
2. Find the distance to the horizon from the
top of Mt. McKinley, Alaska, a height of
6.194 km.
5. 40 feet
7. 150 feet
3. How far is it to the horizon if you are
standing on the ground and your eyes are 2
m above the ground? (Hint: 2 m = 0.002 km)
4. Mauna Kea is an extinct volcano on Hawaii
that is about 4 km tall. You should be able
to see the top of Mauna Kea when you are
how far away?
128 4-6 Estimating Square Roots
8. Short Response The formula
S = 5.5 √
0.8L , where S is the speed in miles
per hour and L is the length of the skid
marks in feet, gives the maximum speeds
that the vehicle was traveling before the
brakes were applied. Find the approximate
range of speed that a vehicle leaving a 200-ft
skid could have been traveling before the
brakes were applied.
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6. 100 feet
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Date
MA.8.A.6.4 Perform operations on
real numbers (including…radicals…
and…irrational numbers) using multistep and real world problems.
Explore It!
Operations with Square Roots
Explore Square Root Relationships
In the following activities, you will explore operations
with square roots and look for relationships between
the numbers involved.
Activity 1
1 How does the grid show that 4 × 9 = 36?
2 Now, take the square root of each part of the equation in Step 1. Is it true that
√
4 × √
9
= √
36 ?
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3 To test whether the relationship holds true in other cases, complete the table.
Do not use a calculator.
m
n
4
9
16
4
9
16
25
4
9
25
√
m
· √
n
2·3=6
√
m·n
√
36 = 6
Try This
Simplify.
1. √
25 · √
36
2. √
25 · 36
3. √9 · √
49
· 49
4. √9
4-7 Operations with Square Roots 129
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Draw Conclusions
5. Describe two ways you can find the product √
9 · √
4.
Activity 2
1 Complete the first 2 rows by hand. Then use a calculator to complete the other
rows. Round square roots to the nearest thousandth.
m
n
√m
____
√
n
m
√__
n
16
4
√
16
4=2
____
= __
2
√
4
16
= √
4=2
√___
4
36
9
100
4
2
6
Try This
Simplify.
√
100
6. _____
√
25
7.
√
144
8. _____
100
√___
25
Draw Conclusions
√
9
64
10. Describe two ways you can find the quotient ____
.
√
√
16
130 4-7 Operations with Square Roots
9.
144
√___
9
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17
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4-7
Date
MA.8.A.6.4 Perform operations on
real numbers (including…radicals…
and…irrational numbers) using multistep and real world problems.
Learn It!
Operations with Square Roots (Student Textbook pp. 172–175)
Lesson Objective
Use the laws of exponents to simplify square roots.
Vocabulary
radical expression
radical symbol
radicand
Example 1
Simplify.
Si
lif
6 + 4√
6
3√
A.
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=(
=
B.
The
) √
6
+
are the same.
Combine like terms.
√
6
2 √
7 + 9 √
3 - 8 √
7
= 9 √
3+
= 9 √
3 +(
= 9 √
3-
Check It Out!
√
7-
-
Property
√
7
) √
7
Combine like terms.
√7
Simplify.
S
1a. -3√
11 + 2 √
11
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1b. -√
14 + 4√
15 + √
14
4-7 Operations with Square Roots 131
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Example 2
Simplify.
Si
lif
A.
√
18 · √8
Multiply the radicands under one
symbol.
·
=√
√
=
=
B.
Simplify.
4 √
20 · √
20
√
=4
√
=4
=
the radicands under one
radical symbol.
Simplify.
= 80
Check It Out!
2a.
S
Simplify.
√
5 · √
20
2b.
√
2·
2 √
2
Simplify
Si lif √
162
16
Method A
Method B
√
·2
√
√
·
=
√
=
√
162 =
132 4-7 Operations with Square Roots
√
9·
√
√
·
=
√
·2
=
√
·
=
√
=
√
162 =
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Example 3
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3. Simplify √
180 .
Example 4
Simplify.
Si
lif
A.
5 - √
125
7√
√
·5
=7 5-
If the radicand has any perfect
squares, factor them out.
= 7√5 -
Simplify.
√
=
√
5
√
5
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B. √
63 + 3√
28
·7 +3
·7
=
√
√
√
√
√
√
·
·
·
=
+3
√ + 3 ·
√
=
√
=
Check It Out!
Factor any
squares out of the radicands.
Simplify.
S
Simplify.
4a. 5√
3 - √
27
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4b. 2 √
24 + 6√
54
4-7 Operations with Square Roots 133
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Date
LA.8.2.2.3 The student will organize
information to show understanding or
relationships…
Operations with Square Roots
Think and Discuss
1. Explain why you can combine terms in the expression √
3 + 6√
3 , but you
cannot combine terms in the expression √
5 + 5 √
2.
2. Show two ways to factor 200 so that each way contains a different perfect
square factor.
3. Get Organized Complete the graphic organizer. Fill in the boxes by giving an
example of a square root that can be simplified and show how to simplify it.
Then give an example of a square root that cannot be simplified.
A Square Root That
Can Be Simplified
Simplified Version
134 4-7 Operations with Square Roots
A Square Root That
Cannot Be Simplified
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Simplifying
Square Roots
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4-7
Date
MA.8.A.6.4 Perform operations on
real numbers (including...radicals
...and…irrational numbers) using
multi-step and real world problems.
Practice It!
Operations with Square Roots
Copyright © by Holt McDougal. All rights reserved.
Simplify.
1. 4√
2 - 2 √
2
2. √
15 + 2 √
15
3. -√
22 - √
22
4. 6 √
13 - 8 √
13 - √
13
5. 7√
23 + √
6 - 2 √
23
6. √
33 - 5 √
33 + √
3
7. √
8 · √
18
8. √
50 · √
8
9. √
6 · √
24
10. 5√
2 · √
200
11. √
19 · √
19
12. √
10 · √
250
13. √
54
14. √
99
15.
16.
√
12
√
44
17. √
7500
18. √
2250
19. 4√
2 + √
18
20. √
300 - √
3
21. -√
1000 + 8 √
10
22. √
108 + 2 √
3 - √
75
23. √
125 - √
245 - √
45
24. 6√
5000 - √
288 + √
32
25. Find the area and perimeter of the rectangle. Write each answer in
simplest form. Show your work.
12 in.
75 in.
4-7 Operations with Square Roots 135
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Date
MA.8.A.6.4 Perform operations on
real numbers (including…radicals…
and…irrational numbers) using
multi-step and real world problems.
Apply It!
Operations with Square Roots
1. Palm Beach County is approximately
shaped like a square with area 2580 mi2.
3. Anne wants to have a square garden with
an area of 200 square feet. Write the length
of each side of the square in simplest
radical form.
PALM BEACH
Atlantic
Ocean
Inland
Florida
BROWARD
2. One student used 5 · 7 = 35 to approximate
√
1250 to the nearest whole number.
Explain how the student arrived at 5 · 7.
5. Gridded Response
The landscaper for a
park needs to know the
area of the park so that
she can buy enough
materials. The plots for
the Library and City
Hall are both squares.
What is the area of
the park in square
meters? (Hint: Find the
dimensions of the City
Hall and Library first.)
Green
Park
City Hall
1200 m2
136 4-7 Operations with Square Roots
Library
300 m2
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Broward County borders Palm Beach
County as shown. Suppose the State of
Florida decided to change the borders of
Broward County to be a square of area
645 mi2. Find the length of the combined
coastline that the two counties share.
Write your answer in simplest radical form.
4. A kicker for the Florida Gators kicked a
football to a height of 128 ft into the air. Use
√
h
the formula, t = ___
4 , where t is the time in
seconds it takes for an object to fall from a
height of h feet. About how long will it take
for the football to come back down to the
field? Write your answer in simplest radical
form.
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4-8
Date
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MA.8.A.6.2 …compare mathematical
expressions involving real numbers and
radical expressions.
The Real Numbers
Explore Real Numbers
Fractions, decimals, whole numbers, natural numbers,
and integers are all rational numbers. In the following
activities, you will explore real numbers that are not
rational numbers.
REMEMBER
• A rational number can be written as a ratio,
a
__
, where a and b are integers and b ≠ 0.
b
Activity 1
Complete the table. Show that each number is a rational number by writing it as a
ratio of two integers __ab .
Number
Ratio __ba
15
15
15 = ___
1
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√
49
√
49
= 7 = __71
-2.37
-237
37
-2.37 = -2___
= _____
100
100
4
___
11
4
___
11
1.278
-5
3
2__
4
√
81
____
13
Try This
1. Show that √
16 is a rational number.
2. Show that -8_35 is a rational number.
3. Show that 6.25 is a rational number.
4-8 The Real Numbers 137
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Activity 2
1 Locate each number on the number line. Draw a point there and write the letter
of the number (A, B, C, etc.) above the point.
(A) 0
(B) √3
(C) -0.75
(D) __73
(E) -√2
5
(F) -__
11
(G) - √
7
(H) 0.423
45
__
(I) - √
5
9
(J) 2__
10
-3
-1
-2
0
1
2
3
An irrational number is a number that can NOT be written as a ratio __ab of two
integers a and b. Square roots of whole numbers that are not perfect squares are
irrational numbers.
2 Which numbers in Step 1 above are irrational numbers?
Try This
Graph each irrational number on the number line.
5. √
6
2
4. √
0
1
2
3
6. √
8
4
7. Explain why every integer is a rational number.
3.5
8. Explain whether ___
is a rational number.
2
9. Give three examples of irrational numbers.
138 4-8 The Real Numbers
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Draw Conclusions
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Class
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4-8
Date
MA.8.A.6.2 …compare mathematical
expressions involving real numbers and
radical expressions
The Real Numbers (Student Edition pp. 176–179)
Lesson Objectives
Determine if a number is rational or irrational
Vocabulary
irrational number
real number
Density Property
Example 1
Write all names that apply to each number.
Copyright © by Holt McDougal. All rights reserved.
A.
√
5
5 is a
number that is not a perfect
.
B. -12.75
-12.75 is a
decimal.
√
16
C. ____
2
√16
4=2
= __
____
2
2
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4-8 The Real Numbers 139
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Write all names that apply to each number.
W
1a. √
9
1b. -__35
1c. 3.724
Example 2
State
the number is rational, irrational, or not a real number. Justify your
St
t if th
answer.
0=
__
3
B.
, because
is a whole number
√
-4
√
-4
; because it is the
of a
negative number
C.
√__49
( __23 )( __23 ) = __49
Check It Out!
2a.
, __23 is rational
State if the number is rational, irrational, or not a real
number. Justify your answer.
√-7
140 4-8 The Real Numbers
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A. __03
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2b. -__60
2c. √
59 + 5
Example 3
Fi
d a reall number between 3__25 and 3__35.
Find
There are many solutions. One solution is halfway between the two
numbers. To find it, add the numbers and divide by 2.
( 3__25 + 3__35 ) ÷ 2 =
=
_____ ÷ 2
5
÷2
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=
3
3
1
5
3
2
5
3
3
5
3
4
5
4
A real number between 3_25 and 3_35 is
.
Check It Out!
3. Find a real number between 5__18 and 5__28.
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4-8 The Real Numbers 141
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Name
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Class
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Date
LA.8.2.2.3 The student will organize
information to show understanding or
relationships…
The Real Numbers
Think and Discuss
1. Explain how rational numbers are related to integers.
2. Tell if a number can be irrational and whole. Explain.
3. Use the Density Property to explain why there are infinitely many real numbers
between 0 and 1.
Rational
Irrational
No
Yes
No
142 4-8 The Real Numbers
Integer
Yes
No
Whole
number
Yes
No
Natural
number
Copyright © by Holt McDougal. All rights reserved.
4. Get Organized Complete the graphic organizer. Fill in the flowchart
by writing each of the following numbers in every box for which the
classification applies: -2, √
3 , 0, 19, _23 .
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Name
Class
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4-8
Date
Practice It!
MA.8.A.6.2 …Compare
mathematical expressions involving
real numbers and radical expressions.
The Real Numbers
Write all names that apply to each number.
1.
36
__
√
4
4. -81
3
2. -__
16
3. √
0.81
5. -7.233
6. √
95
State if the number is rational, irrational, or not a real number.
Copyright © by Holt McDougal. All rights reserved.
7. √
49
21
11. __
0
9
9. ___
8. -√
144
20
12. __
8
√
3
13. √
-100
√
81
10. ____
√
9
14. 8.67
Find a real number between each pair of numbers.
15. 4__25 and 4__35
15
16. 7.25 and __
2
17. __58 and __34
18. Give an example of a rational number between -√
36 and √
36 .
19. Give an example of an irrational number less than 0.
20. Give an example of a number that is not real.
4-8 The Real Numbers 143
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Class
Date
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MA.8.A.6.2 …Compare mathematical
expressions involving real numbers and
radical expressions.
The Real Numbers
1. Twin primes are prime numbers that differ
by 2. Find an irrational number between
twin primes 5 and 7.
Use the number line for 6–8. For each point
on the number line, write a possible rational
and a possible irrational number that it could
represent.
A
2. Rounded to the nearest ten-thousandth,
π = 3.1416. Find a rational number
between 3 and π.
3. One famous irrational number is e.
Rounded to the nearest ten-thousandth
e ≈ 2.7183. Find a rational number that is
between 2 and e.
-5
B
C
0
5
6. A
7. B
8. C
5. Write all the names that apply to any
number that gives the average amount of
rainfall for a week.
144 4-8 The Real Numbers
9. Short Response Explain when the
length of a side of a square would be a
rational number and when it would be an
irrational number.
Copyright © by Holt McDougal. All rights reserved.
4. Perfect numbers are numbers in which
the factors of the number (excluding the
number itself ) add up to the number itself.
For example, the number 6 is a perfect
number because 1 + 2 + 3 = 6. The
number 28 is also a perfect number. Find an
irrational number between 6 and 28.
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4-9
Date
MA.8.G.2.4 Validate and apply
Pythagorean Theorem to find
distances in real world situations or
between points in the coordinate plane.
The Pythagorean Theorem
Explore Right Triangles
In Activity 1, you will explore an interesting relationship
between the side lengths of a right triangle.
REMEMBER
• A right triangle has one right angle.
• An isosceles triangle has two congruent sides.
Activity 1
1 The drawing at the right shows an isosceles right triangle and three
squares. Use grid paper to make a drawing like the one shown. You will
be cutting out the pieces, so be sure your drawing is large enough that
you can easily cut out and work with the pieces.
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2 Cut out the two smaller squares, then cut those
squares in half along a diagonal. Fit the pieces of the
smaller squares on top of the large square.
Try This
1. Compare your results with those of a classmate to confirm this relationship
among the shapes.
Draw Conclusions
2. Describe the relationship between the areas of the small squares
and the large square.
3. How do the side lengths of the triangle relate to the areas of the squares?
4-9 The Pythagorean Theorem 145
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Activity 2
1 Draw a right triangle with legs of 3 units and 4 units on graph paper.
For each leg, draw a square that has a leg as one side. What is the
sum of the areas of these two squares?
3
?
4
2 Measure the length of the hypotenuse using graph paper.
Draw a square with the hypotenuse length as one side of
the square. What is its area?
3
4
3 Compare the sum of the areas you found in Step 1 to the area you found in Step 2.
How are they related?
Repeat Activity 2 for right triangles with legs of the given lengths.
4. 5, 12
5. 6, 8
6. 8, 15
Draw Conclusions
7. How are the square areas next to the sides of the triangle related to the sides
of the triangle?
8. Describe the relationship between the areas of the small squares and the area
of the large square.
146 4-9 The Pythagorean Theorem
Copyright © by Holt McDougal. All rights reserved.
Try This
Learn It!
Explore It!
Name
Summarize It!
Practice It!
Class
Apply It!
4-9
Date
MA.8.G.2.4 Validate and apply
Pythagorean Theorem to find
distances in real world situations or
between points in the coordinate plane.
Learn It!
The Pythagorean Theorem (Student Textbook pp. 180–183)
Lesson Objective
Use the Pythagorean Theorem to solve problems
Vocabulary
Pythagorean Theorem
leg
hypotenuse
Copyright © by Holt McDougal. All rights reserved.
Example 1
Find
i d the
h llength of each hypotenuse to the nearest hundredth.
A.
c
4
5
a2 + b2 = c2
2
Theorem
2
+
= c2
Substitute for a and b.
+
= c2
Simplify powers.
= c2
=c
Solve for c ; c = √
c2 .
≈c
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4-9 The Pythagorean Theorem 147
Learn It!
Explore It!
Summarize It!
Practice It!
Apply It!
Find the length of each hypotenuse to the nearest hundredth.
B. triangle with coordinates (1, -2), (1, 7), (13, -2)
20
The points form a right triangle with a = 9 and b = 12.
2
2
a +b =c
2
2
12
Use the Pythagorean Theorem
4
2
+
-20
= c2
Substitute 9 for a and 12 for b.
= c2
Simplify
-12
-4 O
-4
y
(1, 7)
9
12
4
x
12
20
(1, -2) (13, -2)
-12
+
.
-20
= c2
=c
Check It Out!
Find the
Find the length of each hypotenuse to the nearest hundredth.
F
1a.
1b. triangle with coordinates
(–5, 4), (4, 4), and (4, –6)
c
5
square root.
7
S
l ffor th
Solve
the unknown side in the right triangle to the nearest tenth.
a2 + b2 = c2
25
2
b
7
-
Theorem
2
+ b2 =
Substitute for a and c.
+ b2 =
Simplify powers.
-
b2 =
b=
148 4-9 The Pythagorean Theorem
√
576 = 24
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Example 2
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Learn It!
Summarize It!
Practice It!
Apply It!
Check It Out!
2. Solve for the unknown side in the right triangle.
12
b
4
Example 3
T
Two
airplanes
i l
leave the same airport at the same time. The first plane flies to a
landing strip 350 miles south, while the other plane flies to an airport 725 miles
west. How far apart are the two planes after they land?
a2 + b2 = c2
2
+
Copyright © by Holt McDougal. All rights reserved.
+
Pythagorean Theorem
2
= c2
Substitute for a and b.
= c2
Simplify
.
= c2
≈c
Find the
square root.
The planes are about 805 miles apart after they land.
Check It Out!
3. Two birds leave the same spot at the same time. The first bird flies to his nest
11 miles south, while the other bird flies to his nest 7 miles west. How far apart
are the two birds after they reach their nests?
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4-9 The Pythagorean Theorem 149
Explore It!
4-9
Learn It!
Name
Summarize It!
Practice It!
Apply It!
Class
Summarize It!
Date
LA.8.2.2.3 The student will organize
information to show understanding or
relationships…
The Pythagorean Theorem
Think and Discuss
1. Tell which side of a right triangle is always the longest side.
2. Explain if 2, 3, and 4 cm could be side lengths of a right triangle.
3. Get Organized Complete the graphic organizer. Fill in the boxes by writing the
lengths of the legs and the length of the hypotenuse for the given right triangle.
Then use these lengths to write an equation based on the Pythagorean Theorem.
Lengths of
Legs
5
13
12
Length of
Hypotenuse
150 4-9 The Pythagorean Theorem
Copyright © by Holt McDougal. All rights reserved.
Right
Triangle
Pythagorean
Theorem
Explore It!
Learn It!
Summarize It!
Name
Practice It!
Apply It!
Class
4-9
Date
MA.8.G.2.4 Validate and apply
Pythagorean Theorem to find
distances in real world situations or
between points in the coordinate plane.
Practice It!
The Pythagorean Theorem
Find the length of each hypotenuse. Round to the nearest hundredth if necessary.
1.
2.
9
3.
36
24
27
12
10
4.
5.
8
6.
5
7
15
9
6
Solve for the unknown side in each right triangle. Round to the nearest tenth if necessary.
7.
8.
Copyright © by Holt McDougal. All rights reserved.
50
9.
14
25
9
3.6
2.7
10.
20
11.
28
12.
15
30
21
14
13. A rectangular swimming pool in a park is 60 feet long and 25 feet
wide. Marsha swims from one corner of the pool to the opposite
corner and back 10 times. How many feet does she swim?
14. To meet federal guidelines, a wheelchair ramp that is constructed
to rise 1 foot off the ground must extend 12 feet along the ground.
How long will the ramp be? Round your answer to the nearest
hundredth.
4-9 The Pythagorean Theorem 151
Explore It!
4-9
Learn It!
Name
Apply It!
Summarize It!
Practice It!
Class
Apply It!
Date
MA.8.G.2.4 Validate and apply
Pythagorean Theorem to find
distances in real world situations or
between points in the coordinate plane.
The Pythagorean Theorem
2. A 12-foot ladder is resting against a wall.
The base of the ladder is 2.5 feet from the
base of the wall. How high up the wall will
the ladder reach? Round your answer to the
nearest tenth.
4. A television screen measures approximately
15.5 in. high and 19.5 in. wide. The television
is advertised by giving the approximate
length of the diagonal of its screen, to the
nearest whole inch. What is the advertised
size of this television? Show your work.
152 4-9 The Pythagorean Theorem
a. Find the length of the diagonal of the
entire field, including the end zones.
b. How long would it take a player running
at 22 ft per second to run the length of
the diagonal?
6. The base-path of a baseball diamond forms
a square. If it is 90 ft from home to first, how
far does the catcher have to throw to catch
someone stealing second base? Round your
answer to the nearest tenth.
7. Short Response
Hennrick is making
a kite. The lengths
of the rods for the
frame are shown in
the diagram. Find the
perimeter of the kite
to the nearest tenth.
Show your work.
18 in.
Copyright © by Holt McDougal. All rights reserved.
3. The glass for a picture window is 8 feet by
10 feet. The door it must pass through is 3
feet by 7.5 feet. Will the glass fit through the
door? Justify your answer.
5. A football field is 100 yards with 10 yards at
each end for the end zones. The field is 45
yards wide. Use this information for 5 and 6.
Round your answer to the nearest tenth.
27 in.
1. A 10-m tall utility pole is supported by two
guy wires. Each guy wire reaches from the
top of the poll down to the ground 3 meters
away from the base of the pole. How many
meters of wire are needed for the two guy
wires? Round your answer to the nearest
tenth.
Explore It!
Learn It!
Summarize It!
Name
Practice It!
Apply It!
Class
4-10
Date
MA.8.G.2.4 Validate and apply
Pythagorean Theorem to find
distances in real world situations or
between points in the coordinate plane.
Explore It!
Applying the Pythagorean Theorem and Its Converse
Explore the Pythagorean Theorem
The Pythagorean Theorem states that for any right
triangle with legs a and b and hypotenuse c,
a2 + b 2 = c 2 .
c
a
In the following activity, you will use the Pythagorean
Theorem to find different right-triangle side lengths
that are all integers.
b
a2 + b2 = c2
Activity
1 Complete the table below. Then try choosing your own values for m and n for the last
two rows.
Use these rules for choosing m and n:
1. m and n should be positive whole numbers, with m > n.
2. One number should be odd and the other should be even.
Copyright © by Holt McDougal. All rights reserved.
3. m and n should not have any common factors.
m
n
a = m2 - n2
b = 2mn
c = m2 + n2
Does a2 + b2 = c 2?
2
1
3
4
5
yes
3
2
5
4
1
4
3
5
2
5
4
6
1
4-10 Applying the Pythagorean Theorem and Its Converse 153
Explore It!
Learn It!
Summarize It!
Practice It!
2 The figure at the right shows that if the values of a, b, and c in the first
row of the table (3, 4, and 5) are used to construct the three sides of a
triangle, it will be a right triangle.
Apply It!
5
3
Choose another row of the table. Use the values of a, b, and c in that row
to construct the three sides of a triangle below.
4
3 Describe your results.
Try This
Use the given values of m and n to calculate a, b, and c.
1. m = 6, n = 5
2. m = 7, n = 4
3. m = 7, n = 2
a=
a=
b=
b=
b=
c=
c=
c=
Draw Conclusions
4. Three numbers a, b, and c have the property that a2 + b2 = c2. Describe a
triangle that has sides of length a units, b units, and c units.
154 4-10 Applying the Pythagorean Theorem and Its Converse
Copyright © by Holt McDougal. All rights reserved.
a=
Learn It!
Explore It!
Summarize It!
Name
Practice It!
Apply It!
Class
4-10
Date
MA.8.G.2.4 Validate and apply
Pythagorean Theorem to find distances
in real world situations or between
points in the coordinate plane.
Learn It!
Applying the Pythagorean Theorem and Its Converse (Student Textbook pp. 184–187)
Lesson Objective
Use the Distance Formula and the Pythagorean Theorem and its converse to
solve problems
Example 1
Whatt iis th
Wh
the d
diagonal length of the rectangular projector screen below?
7 ft
Copyright © by Holt McDougal. All rights reserved.
3 ft
+
= c2
Use the
+
= c2
Simplify.
.
= c2
Add.
=c
Take the
of both sides.
≈c
Find the
.
The diagonal length is about
feet.
Check It Out!
1. A square garden has a side length of 10 meters. What is the length of the
diagonal of the garden, to the nearest hundredth?
4-10 Applying the Pythagorean Theorem and Its Converse 155
Learn It!
Explore It!
Summarize It!
Practice It!
Apply It!
Example 2
Fi
d th
t
Find
the di
distances
between the points to the nearest tenth.
y
2
L x
J
-4
O
-2
2
4
-2
M
K
-4
A. J and K
J(-4, 0) and K(0, -3)
Let J be (x2, y2) and K be (x1, y1).
d = √
(x2 - x1)2 + (y2 - y1)2
Use the
√
(
) +(
)
= √(
) +
2
=
2
2
.
Substitute.
2
Subtract.
√
+
=
=√
=
Simplify powers.
Add, then take the square root.
units.
B. L and M
L(4,0) and M(5, -3)
Let L be (x2, y2) and M be (x1, y1).
d = √
(x2 - x1)2 + (y2 - y1)2
Use the
√
(
) +(
)
+
=√
+
=√
≈
=√
2
=
2
2
The distance between L and M is about
2
Substitute.
Subtract.
Simplify powers.
Add, then take the square root.
units.
156 4-10 Applying the Pythagorean Theorem and Its Converse
.
Copyright © by Holt McDougal. All rights reserved.
The distance between J and K is
Explore It!
Learn It!
Check It Out!
Summarize It!
Practice It!
Apply It!
Find the distance between the points to the nearest tenth.
2a. K and L
2
2b. J and M
Example 3
T
ll whether
h th the given side lengths form a right triangle.
Tell
A. 9, 12, 15
a2 + b2 = c2
2
2
Compare a2 + b2 to c2.
2
+
Substitute.
+
Simplify.
✔
=
The side lengths
Add.
a right triangle.
B. 8, 10, 13
a 2 + b2 = c 2
Copyright © by Holt McDougal. All rights reserved.
2
Compare a2 + b2 to c2.
2
+
Substitute.
+
Simplify.
≠
The side lengths
Check It Out!
3a. 8, 11, 13
3
✘
Add.
form a right triangle.
Tell whether the given side lengths form a right triangle.
3b. 18, 24, 30
4-10 Applying the Pythagorean Theorem and Its Converse 157
4-10
Explore It!
Summarize It!
Learn It!
Name
Practice It!
Class
Summarize It!
Apply It!
Date
LA.8.2.2.3 The student will organize
information to show understanding or
relationships…
Applying the Pythagorean Theorem and Its Converse
Think and Discuss
1. Make a conjecture about whether doubling the side lengths of a right triangle
makes another right triangle.
2. Get Organized Complete the graphic organizer. Fill in the boxes by writing
the statement of the Distance Formula. Then give an example of how to use the
formula by giving coordinates of point A and point B and showing how to find the
distance between the points.
Distance
Formula
Statement
Point A
Point B
158 4-10 Applying the Pythagorean Theorem and Its Converse
Distance
Between A and B
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Example
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Practice It!
Summarize It!
Name
Apply It!
Class
4-10
Date
MA.8.G.2.4 Validate and apply
Pythagorean Theorem to find
distances in real world situations or
between points in the coordinate plane.
Practice It!
Applying the Pythagorean Theorem and Its Converse
Determine whether each triangle is a right triangle.
1.
2.
25
26
16
3.
12
50
14
48
10
10
4.
5.
10 ft
20 mm
12 mm
7 ft
16 mm
6.
7.
30 m
10 m
17 in.
8 in.
15 in.
25 m
Copyright © by Holt McDougal. All rights reserved.
6 ft
Tell whether the given side lengths form a right triangle.
8. 7, 10, 12
9. 15, 20, 25
10. 9, 11, 14
11. 5, 7, 12
12. A basketball court is 94 feet long and 50 feet wide. What is the
length of a diagonal of the basketball court, to the nearest tenth?
Find the distance between each pair of points.
13. (3, 6) and (1, 2)
14. (-5, -2) and (-8, 3)
15. (-5, 3) and (5, 5)
16. (3, 4) and (-4, -1)
4-10 Applying the Pythagorean Theorem and Its Converse 159
Explore It!
4-10
Learn It!
Name
Apply It!
Summarize It!
Practice It!
Class
Apply It!
Date
MA.8.G.2.4 Validate and apply
Pythagorean Theorem to find
distances in real world situations or
between points in the coordinate plane.
Applying the Pythagorean Theorem and Its Converse
1. Federal guidelines require that a wheelchair
ramp must extend at least 12 units along
the ground for every 1 unit off the ground
that it rises vertically. What is the minimum
slanted length of a wheelchair ramp that
reaches the top of a 5-ft-high staircase?
Explain.
On a map, each unit on the grid represents a
mile. Use the information for 2 and 3.
3. A post office is located at (2, 2). Find a point
that is 13 miles from the location of the post
office. Show your work.
5. Jorge wants to build a support in the shape
of a right triangle. He has one 9-foot board
and one 4-foot board. What are the two
possible lengths he needs for the third
board? Round your answer to the nearest
hundredth.
6. Extended Response Tony needs to
use a ladder to get onto the roof of an 11-ft
house. His ladder is 14 ft long. According
to safety regulations, the base of the ladder
should be placed 6 ft from the base of the
house, and the ladder should extend at least
1 ft over the roofline. Can Tony safely use
his ladder to climb onto the roof? Justify
your answer.
160 4-10 Applying the Pythagorean Theorem and Its Converse
Copyright © by Holt McDougal. All rights reserved.
2. One city is located at (4, 8) and another city
is located at (6, –12) on the grid. How many
miles apart are the two cities? Round to the
nearest tenth of a mile. Show your work.
4. Linda made triangular flags for the spirit
club to wave. Each flag was a right triangle.
One side was 1.5 feet long and another
side was 2.2 feet long. She used fringed
trim along the longest side of the each flag.
What was the length of fringed trim that she
sewed to each flag? Round to the nearest
tenth of a foot.
Name
Class
Got It?
4-6
Date
THROUGH
4-10
Ready to Go On?
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Quiz for Lessons 4-6 through 4-10
4-6
Estimating Square Roots (Student Textbook pp. 168–171)
Each square root is between two consecutive integers. Name the integers.
72
1. -√
2. √
200
3. -√
340
4. √
610
5. The area of a chess board is 110 square inches.
Find the length of one side of the board to
the nearest hundredth.
4-7
Operations with Square Roots (Student Textbook pp. 172–175)
Simplify.
7 - √
7
6. 7√
9. √
288 + √
2
4-8
7. √
450 · √
2
10.
8. √
108
√
2 - √
128
11. 10√
20 + 5√
320
The Real Numbers (Student Textbook pp. 176–179)
Write all names that apply to each number.
Copyright © by Holt McDougal. All rights reserved.
12.
√
12
13. 0.15
14. √
1600
15. Give an example of an irrational number that is less than -5.
4-9
The Pythagorean Theorem (Student Textbook pp. 180–183)
Find the missing length for each right triangle. Round your answer to the
nearest tenth, if necessary.
16. a = 3, b = 6, c =
17. a =
, b = 24, c = 25
4-10
Applying the Pythagorean Theorem and Its Converse
(Student Textbook pp. 184–187)
Find the distance between the points. Round to the nearest tenth, if necessary.
18. (3, 2) and (11, 8)
19. (-1, -1) and (-3, 6)
Tell whether the given side lengths form a right triangle.
20. 7, 9, 11
21. 8, 14, 17
Chapter 4 Exponents and Roots 161
4-6
THROUGH
4-10
Name
Class
Date
Connect It!
MA.8.A.6.2, MA.8.A.6.3,
MA.8.A.6.4, MA.8.G.2.4
Connect the concepts of Lessons 4-6 through 4-10
Spiraling Out of Control
1. On a sheet of graph paper, draw an isosceles right triangle (Triangle A) with
legs 1 unit long. Find the length of the hypotenuse and record it in the table.
A1
1
2. Draw a new isosceles right triangle (Triangle B) so that one of its legs is the
hypotenuse of the first triangle, as shown. Find and record the length of the
legs and hypotenuse of Triangle B.
B
A
3. Continue to draw new isosceles right triangles in this way and record the
length of the legs and hypotenuse for each triangle. Simplify any expressions
with square roots.
C
4. When you have drawn 8 triangles (Triangles A through H), look for patterns
in your completed table. Without drawing any new triangles, predict the
lengths of the legs of Triangles I, J, K, and L.
Triangle
A
Legs
1
B
C
D
E
F
G
B
A
H
Hypotenuse
1. Find pairs of expressions in the figure that have
the same value. When you find a matching pair,
cross out the expressions.
2. When you have crossed out all the matching
pairs, the leftover expression will tell you
the number of pounds of turkey the average
American eats each year. Estimate this value to
the nearest tenth.
144
40 + 90
22 + 42
100 + 4
82 + 62
2 5
102
-12
12
250
Think About The Puzzler
3. Explain how you found one of the matching pairs.
162 Chapter 4 Exponents and Roots
172
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Serving Leftovers
FLORIDA
Name
Class
Study It!
Vocabulary
CHAPTER
Date
4
Multi-Language
Glossary
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(Student Textbook page references)
Density Property . . . . . . . . (176)
principal square root . . . . (164)
radicand . . . . . . . . . . . . . . . (172)
hypotenuse . . . . . . . . . . . . . (180)
Pythagorean Theorem. . . (180)
real number . . . . . . . . . . . . (176)
irrational number. . . . . . . (176)
radical expression . . . . . . (172)
scientific notation. . . . . . . (152)
leg . . . . . . . . . . . . . . . . . . . . . (180)
radical symbol . . . . . . . . . . (172)
square root . . . . . . . . . . . . . (164)
perfect square . . . . . . . . . . (164)
Complete the sentences below with vocabulary words from the list above.
1. A(n)
is a number that cannot be written as a fraction.
2.
is a short-hand way of writing extremely large or extremely
small numbers.
3. The
states that the sum of the squares of the
Copyright © by Holt McDougal. All rights reserved.
of a right triangle is equal to the square of the
.
Lesson 4-2
IInteger Exponents (Student Textbook pp. 148–151)
Simplify.
( -3 )-2
MA.8.A.6.1
20
20 = 1
1 = __
1
(-3)-2 = _____
2
9
( -3 )
Simplify.
4. 5-3
5. ( -4 )-3
6. 11-1
7. 10-4
8. -6-2
9. ( 9 - 7 )-3
10. ( 6 - 9 )-3
11. 4-1 + ( 5 - 7 )-2
Lesson 4-3
12. 3-2 · 2-3 · 90
Scientific Notation (Student Textbook pp. 152–156)
S
Write in scientific notation.
Write in standard notation.
4
3.58 × 10
3.58 × 10,000
35,800
MA.8.A.6.1
-4
3.58 × 10
1
3.58 × ______
10,000
3.58 × 0.0001
0.000358
Lesson Tutorial Videos @ thinkcentral.com
0.000007
7 × 0.000001
1
7 × ________
1,000,000
62,500
6.25 × 10,000
6.25 × 104
7 × 10-6
Chapter 4 Exponents and Roots 163
Write in standard notation.
14. 1.62 × 10-3
13. 1.62 × 103
15. 9.1 × 10-5
Write in scientific notation.
16. 385
17. 0.04
Lesson 4-4
18. 0.000000008
19. 73,000,000
LLaws of Exponents (Student Textbook pp. 157–161)
MA.8.A.6.3
Simplify. Write as one power.
25 · 23
25 + 3
28
109
___
102
Add exponents.
109 - 2
107
( 42 )3
42 · 3
46
Subtract
exponents.
Multiply
exponents.
Simplify. Write as one power.
20. 42 · 45
0
24. 5 · 5
m
3 )4
3
4-5
23. ___
-3
m7
22. ___
2
21. p · p3
4
6
25. ( 2
26. y ÷ y
5
27. ( 23 · 2 )
2
28. A hummingbird weighs about 1.5 × 10-2 pound.
Write the weight of 50 hummingbirds in scientific notation.
Lesson 4-5
Squares and Square Roots (Student Textbook pp. 164–167)
S
Find the two square roots of 400.
Simplify each expression.
64
___
√
16
√
49 - 10
7 - 10
-3
√
64
____
= __84 = 2
√
16
√
49 · 4
√
49 · √
4
7·2
14
Find the two square roots of each number.
29. 16
30. 900
31. 676
√
100
33. _____
20
34. √
34
Simplify each expression.
32. √
4 + 21
Lesson 4-6
Estimating Square Roots (Student Textbook pp. 168–171)
E
MA.8.A.6.2
Approximate √
359 to the nearest hundredth.
√
324 < √
359
< √
361
Step 1: 18 < √
359 < 19 The whole number part is 18.
35
__
359
324
35; 361
324
Step 2: 37 ≈ 0.95
√
359 is approximately 18.95.
164 Chapter 4 Exponents and Roots
37
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20 · 20 = 400
( -20 ) · ( -20 ) = 400
The square roots are 20 and -20.
MA.8.A.6.2
Each square root is between two consecutive integers. Name the integers.
35. -√
43
36. √
1000
37.
√
75
38. Approximate √
105 to the nearest hundredth.
Lesson 4-7
Operations with Square Roots (Student Textbook pp. 172–175)
O
Simplify √
192 - 2 √
3.
√
64 · 3 - 2 √
3
Find perfect square factors of the radicand.
8 √
3 - 2 √
3
Simplify.
6 √
3
Combine like terms.
MA.8.A.6.4,
MA.8.A.6.2
Simplify.
39. -2 √
11 + √
11
40. 3 √
10 · √
40
42. -√
250 + 6 √
10
43.
Lesson 4-8
41. √
360
√
2 · √
4050
44. √
32 - 6 √
2
The Real Numbers (Student Textbook pp. 176–179)
T
MA.8.A.6.2
State if the number is rational, irrational, or not a real number.
-√
2 irrational
The decimal equivalent does not repeat or end.
√
-4 not real
Square root of a negative number
State if the number is rational, irrational, or not a real number.
Copyright © by Holt McDougal. All rights reserved.
45.
Lesson 4-9
0
47. ___
-4
46. √
-16
√
122
The Pythagorean Theorem (Student Textbook pp. 180–183)
T
MA.8.G.2.4,
MA.8.A.6.4
Find the length of side b in the right triangle where a = 8 and c = 17.
82 + b2 = 172 Use the Pythagorean Theorem: a2 + b2
64 + b2 = 289
b2 = 289 - 64 = 225 → b = √
225 = 15
c2
Find the side length in each right triangle.
48. If a = 6 and b = 8, find c.
Lesson 4-10
49. If b = 24 and c = 26, find a.
Applying the Pythagorean Theorem and Its Converse
A
((Student Textbook pp. 184–187)
Find the distance between (3, 7) and (–5, 6) to the nearest tenth.
MA.8.G.2.4,
MA.8.A.6.4
√
( -5 - 3 )2 + ( 6 - 7 )2 = √
( -8 )2 + ( -1 )2 = √
64 + 1 = √
65 ≈ 8.1
Find the distance between each pair of points, to the nearest tenth.
50. ( 1, 4 ), ( 2, 7 )
51. ( –2, 3 ), ( 6, 9 )
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52. ( 5, –2 ),( –4, 10 )
Chapter 4 Exponents and Roots 165
Name
Class
Write About It!
Date
LA.8.3.1.2 The student will prewrite
by making a plan for writing that
addresses purpose, audience, main
idea, logical sequence, and time frame for
completion.
Think and Discuss
Answer these questions to summarize the important concepts from Chapter 4
in your own words.
1. Explain how to evaluate 36.
2. Explain the difference between 3.56 × 108 and 3.56 × 10-8.
3. Explain why 81 has two square roots.
4. Explain how to estimate √
60 .
__
6. Explain why 0.3 is a rational number.
Before the Test
I need answers to these questions:
166 Chapter 4 Exponents and Roots
Copyright © by Holt McDougal. All rights reserved.
5. Show three different ways to express the exact value of √
2160 .