Download Powers and Exponents

LESSON
Page 1 of 6
1.2
Powers and Exponents
Now
BEFORE
You multiplied whole
numbers and decimals.
Vocabulary
power, p. 10
base, p. 10
exponent, p. 10
WHY?
You’ll use powers to describe
repeated multiplication.
So you can find the total number
of e-mails sent, as in Ex. 29.
A power is the result of a repeated multiplication of the same factor.
For example, the number 125 is a power because 125 5 p 5 p 5. A power
can be written in a form that has two parts: a number called the base
and a number called the exponent . The exponent shows the number of
times the base is used as a factor.
exponent
The base 5 is used
as a factor 3 times.
53 5 p 5 p 5
base
power
factors
The table shows how to read and write powers. Numbers raised to the
first power, such as 121, are usually written without the exponent.
Power
121
(0.5)2
43
84
In words
12 to the first power
0.5 to the second power, or 0.5 squared
4 to the third power, or 4 cubed
8 to the fourth power
Example 1
Note Worthy
Write additional examples of
your own in your notebook. For
each product, identify the base
and the exponent, then write
the product using an exponent.
Value
121 12
(0.5)(0.5) 0.25
4 p 4 p 4 64
8 p 8 p 8 p 8 4096
Using Exponents
Write the product using an exponent.
a.
b.
c.
d.
13 p 13 p 13 p 13 134
The base 13 is used as a factor 4 times.
(0.2)(0.2)(0.2) (0.2)3
The base 0.2 is used as a factor 3 times.
6
npnpnpnpnpnn
tptptptptt
5
The base n is used as a factor 6 times.
The base t is used as a factor 5 times.
Checkpoint
Write the product using an exponent.
1. 10 p 10 p 10
3. x p x p x p x
2. (4.3)(4.3)
2
3
4
4. Critical Thinking Evaluate each power: 0 , 0 , 0 . Use your results to
write a rule for the value of 0 raised to any nonzero whole number
exponent.
10
Chapter 1
Variables, Expressions, and Integers
Page 2 of 6
Example 2
Evaluating Powers with Variables
Evaluate the expression x 4 when x 0.5.
x 4 (0.5)4
Substitute 0.5 for x.
(0.5)(0.5)(0.5)(0.5)
Use 0.5 as a factor 4 times.
0.0625
Multiply.
Checkpoint
Evaluate the expression when m 3.
5. m2
Review Help
6. m3
7. m4
8. m5
Using Formulas A formula describes a relationship between quantities.
Some formulas involve powers. For example, you can use a formula to
find the area of a square or the volume of a cube.
For help with area and volume,
see p. 789.
s
s
s
s
s
Area A of a square
As
Volume V of a cube
2
V s3
Area is measured in square units, such as square feet (ft2) or square
centimeters (cm2). Volume is measured in cubic units, such as cubic
inches (in.3) or cubic meters (m3).
Example 3
Using Powers in Formulas
Ice Sculpture An artist uses a cubeshaped block of ice to make an ice
sculpture for a competition. Find the
volume of the block of ice.
20 in.
Solution
Use the formula for the volume of a cube.
Vs
3
Write the formula.
3
In the
Real World
Ice Sculpture A cubic inch of
ice weighs about 0.03 pound. If
the artist carves away about
2000 cubic inches from the
block in Example 3, about how
much does the resulting
sculpture weigh?
20 in.
(20)
Substitute 20 for s.
8000
Evaluate power.
20 in.
Answer The volume of the block of ice is 8000 cubic inches.
Checkpoint
Find the area of a square with the given side length.
9. 9 meters
10. 11 inches
Lesson 1.2
11. 1.5 centimeters
Powers and Exponents
11
Page 3 of 6
1.2
Exercises
INTERNET
More Practice, p. 803
CLASSZONE.COM
eWorkbook Plus
Guided Practice
Vocabulary Check
1. Identify the base and the exponent in the expression 135.
2. How are the expressions 34 and 43 different?
Skill Check
Write the power in words and as a repeated multiplication. Then evaluate
the power.
3. 122
4. (0.3)3
5. (1.2)3
6. 54
Evaluate the expression when k 6.
7. k 2
8. k 3
9. k 4
10. k 5
11. Gift Box A gift box has the shape of a cube with an edge length of
14 inches. Find the volume of the box.
12. Error Analysis Describe and correct
the error in writing 23 as a repeated
multiplication.
23 3 p 3
Practice and Problem Solving
Homework Help
Example
1
2
3
Exercises
13–29
30–33
36
CLASSZONE.COM
13. 32 p 32
14. 11 p 11 p 11
17. (5.6)(5.6)(5.6) 18. (1.7)(1.7)
15. 6 p 6 p 6 p 6 p 6 16. 2 p 2 p 2 p 2
19. z p z p z
20. n p n p n p n
Write the power in words and as a repeated multiplication. Then
evaluate the power.
Online Resources
• More Examples
• eTutorial Plus
Write the product using an exponent.
21. 83
22. 25
23. 106
24. 123
25. 93
26. 44
27. (0.2)2
28. (0.6)4
29. Extended Problem Solving You
send an e-mail to 4 friends. Each
friend sends the e-mail to 4 more
friends. Each of those friends sends
it to 4 friends, and so on.
a. Copy and complete the table.
Stage
1
2
3
4
E-mails sent,
as a power
41
42
?
?
Value of
power
4
16
?
?
b. Calculate Find the number of
e-mails sent at stage 9.
c.
12
Chapter 1
Estimate Estimate the stage at which more than 1,000,000
e-mails will be sent. Use a calculator to check your estimate.
Variables, Expressions, and Integers
Page 4 of 6
Evaluate the expression when n 7 and when n 0.4.
30. n2
34.
31. n3
32. n4
33. n5
Writing The square of a number is the second power of the number.
The cube of a number is the third power of the number. Explain why
these names are reasonable.
35. Critical Thinking Explain why 1 raised to any power is equal to 1.
36. Aquariums An aquarium has a square base with a side length of
15 inches. You fill the aquarium with water to a height of 15 inches.
a. Find the volume of the water in the aquarium.
b. A cubic inch of water weighs approximately 0.036 pound. Find the
approximate weight of the water in the aquarium.
37. Patterns The table shows sums of odd
numbers.
Sum of first n
odd numbers
1
134
135?
1 3 5 7 ?
?
n
a. Copy and complete the table. Identify any
pattern that you see.
1
2
3
4
5
b. Write a variable expression for the sum of
the first n odd numbers.
c. Use your expression from part (b) to find
the sum of the first 100 odd numbers.
38. Challenge Find values of x, y, and z so that each of the expressions x 2,
y 3, and z 6 has a value of 64.
Find the product or quotient. (pp. 775, 776)
39. (2.5)(7.1)
41. 1.2 2.4
40. (2.3)(8.4)
43. Olympics The bar graph shows the
Olympic Gold Medals in 2000
50
40
30
20
10
0
28
tra
Au
s
Evaluate the expression when x 15. (Lesson 1.1)
44. x 4
45. 200 x
46. x 11
Standardized Test
Practice
32
39
16
lia
Medals
number of gold medals won by the
four countries with the most gold
medals in the 2000 Olympic
Summer Games. How many gold
medals did the four countries win
in all? (p. 781)
42. 5.2 1.25
Ch
ina
Ru
ss
ia
Un
St ite
at d
es
Mixed Review
47. 3x
48. Multiple Choice Which expression has a value of 81?
A. 43
B. 34
C. 28
49. Short Response Compare each
number in the top row of the table with
the number below it. Describe any
pattern you see. Complete the table
with a variable expression involving n.
Lesson 1.2
D. 273
1
2
3
4
...
n
1
8
27
64
...
?
Powers and Exponents
13
Page 5 of 6
A Problem Solving Plan
Review
this
topic in preparation
for solving problems
in Lesson 1.3. For a
review of problem
solving strategies,
see pp. 795–802.
A Problem Solving Plan
You can use the following 4-step plan to solve a problem.
1
Read and Understand Read the problem carefully. Identify the question
and any important information.
2
Make a Plan Decide on a problem solving strategy.
3
Solve the Problem Use the problem solving strategy to answer the
question.
4
Look Back Check that your answer is reasonable.
Reading and Planning
Example You plan to ship 5 books to a friend. The table shows the masses
of the books. Is it possible to ship the books in 2 boxes, each with a mass of
6 kilograms or less? Explain.
Book
Mass (kg)
A
B
C
D
E
1.4
2.1
3.8
1.9
2.5
Read and Understand
What do you know?
The mass of each box must be 6 kilograms or less.
The table gives the mass of each book.
What do you want to find out?
Is it possible to put the books in 2 boxes so that each box has a mass of
6 kilograms or less?
Make a Plan
How can you relate what you know to what you want to find out?
Check that the total mass of the books doesn’t exceed 12 kilograms. If it does,
you can’t divide the books as you want.
Use the strategy guess, check, and revise to choose books for each box.
14
Chapter 1
Variables, Expressions, and Integers
Page 6 of 6
Solving and Looking Back
To solve the problem from page 14 about shipping books, carry out the
plan. Then check the answer.
Solve the Problem
The total mass of the books is 1.4 2.1 3.8 1.9 2.5 11.7 kg, so it may
be possible to ship the books as you want.
Now use the strategy guess, check, and revise.
Put the 3 lightest books in one box and the
2 heaviest in the other. The mass of the
second box is more than 6 kg.
Try switching books B and E. The mass of
each box is less than 6 kg.
Books
A, D, B
C, E
Total mass (kg)
1.4 1.9 2.1 5.4
3.8 2.5 6.3
Books
A, D, E
C, B
Total mass (kg)
1.4 1.9 2.5 5.8
3.8 2.1 5.9
Answer It is possible to ship the books in 2 boxes, each with a mass of
6 kg or less. Put books A, D, and E in one box and books C and B in the other.
Look Back
It makes sense that the box with 3 books contains the 2 lightest books and
the box with 2 books contains the heaviest book. So, the answer is
reasonable.
Checkpoint
Test
your
knowledge of the
problem solving plan
by solving these
problems.
1. Pool Schedules A community pool offers
3 swim sessions each Saturday morning.
Each session lasts 35 minutes, with
10 minutes between sessions. The final
session ends at 11:05 A.M. At what time
does the first session begin?
2. Theater Seating The center section of a
theater has 10 rows. There are 41 seats in
row 10, 38 seats in row 9, 35 seats in row 8,
and so on. How many seats are in row 1?
3. Movie Marathon You have invited friends over
to watch movies. You have rented 4 movies:
one action, one science fiction, one comedy,
and one animated. In how many different
orders can you watch the movies?
Student Reference
A Problem Solving Plan
15