10.1 Exponents
How can you use exponents to
write numbers?
The expression 35 is called a power. The base is 3. The exponent is 5.
base
1
35
exponent
ACTIVITY: Using Exponent Notation
Work with a partner.
a. Copy and complete the table.
Power
Repeated Multiplication Form
Value
(−3)1
−3
−3
(−3)2
(−3) (−3)
⋅
9
(−3)3
(−3)4
(−3)5
(−3)6
(−3)7
b. REPEATED REASONING Describe what is meant by the expression (−3)n.
How can you find the value of (−3)n ?
2
Exponents
In this lesson, you will
● write expressions using
integer exponents.
● evaluate expressions
involving integer
exponents.
ACTIVITY: Using Exponent Notation
attion
Work with a partner.
a. The cube at the right has $3 in each
of its small cubes. Write a power
that represents the total amount
of money in the large cube.
b. Evaluate the power to find the
total amount of money in the
large cube.
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3
ACTIVITY: Writing Powers as Whole Numbers
Work with a partner. Write each distance as a whole number. Which numbers
do you know how to write in words? For instance, in words, 103 is equal to
one thousand.
a. 1026 meters:
b. 1021 meters:
diameter of
observable
universe
diameter of
Milky Way galaxy
c. 1016 meters:
d. 107 meters:
diameter of
solar system
diameter of
Earth
e. 106 meters:
f. 105 meters:
Math
Practice
Analyze Givens
What information is
given in the poem?
What are you trying
to find?
h
4
width of
Lake Erie
dt
Wi
length of
Lake Erie
shoreline
ACTIVITY: Writing a Power
Work with a partner. Write the numbers
of kits, cats, sacks, and wives as powers.
As I was going to St. Ives
I met a man with seven wives
Each wife had seven sacks
Each sack had seven cats
Each cat had seven kits
Kits, cats, sacks, wives
How many were going to St. Ives?
Nursery Rhyme, 1730
5. IN YOUR OWN WORDS How can you use exponents to write numbers?
Give some examples of how exponents are used in real life.
Use what you learned about exponents to complete Exercises 3–5
on page 414.
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10.1 Lesson
Lesson Tutorials
A power is a product of repeated factors. The base of a power is the
common factor. The exponent of a power indicates the number of times
the base is used as a factor.
Key Vocabulary
power, p. 412
base, p. 412
exponent, p. 412
exponent
base
⋅ ⋅ ⋅ ⋅
()
1 5 1 1 1 1 1
=— — — — —
2
2 2 2 2 2
—
1
2
— is used as a factor 5 times.
power
EXAMPLE
Writing Expressions Using Exponents
1
Write each product using exponents.
⋅
⋅
a. (−7) (−7) (−7)
Study Tip
Because −7 is used as a factor 3 times, its exponent is 3.
Use parentheses to
write powers with
negative bases.
⋅
π ⋅π ⋅r ⋅r ⋅r
⋅
So, (−7) (−7) (−7) = (−7)3.
b.
Because π is used as a factor 2 times, its exponent is 2. Because r is
used as a factor 3 times, its exponent is 3.
⋅ ⋅⋅⋅
So, π π r r r = π 2r 3.
Write the product using exponents.
1
4
Exercises 3–10
⋅ 14 ⋅ 14 ⋅ 14 ⋅ 14
1. — — — — —
EXAMPLE
2.
⋅ ⋅ ⋅ ⋅⋅
0.3 0.3 0.3 0.3 x x
Evaluating Expressions
2
Evaluate each expression.
a. (−2)4
⋅
⋅
⋅
(−2)4 = (−2) (−2) (−2) (−2)
The base is −2.
b. −24
= 16
Simplify.
⋅ ⋅ ⋅
−24 = −(2 2 2 2)
The base is 2.
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Write as repeated multiplication.
= −16
Write as repeated multiplication.
Simplify.
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EXAMPLE
3
Using Order of Operations
Evaluate each expression.
⋅
⋅
a. 3 + 2 34 = 3 + 2 81
Evaluate the power.
= 3 + 162
Multiply.
= 165
Add.
b. 33 − 82 ÷ 2 = 27 − 64 ÷ 2
Evaluate the powers.
= 27 − 32
Divide.
= −5
Subtract.
Evaluate the expression.
Exercises 11–16
and 21–26
EXAMPLE
3. −54
4
4.
( )
1 3
6
−—
5.
∣ −33 ÷ 27 ∣
⋅
6. 9 − 25 0.5
Real-Life Application
In sphering, a person is secured inside a small, hollow sphere that
is surrounded by a larger sphere. The space between the spheres is
inflated with air. What is the volume of the inflated space?
You can find the radius of each sphere by dividing each diameter
given in the diagram by 2.
Outer Sphere
2m 3m
4
3
V = —πr3
()
π( )
4
3
4
3
9
2
= —π
4
3
V = —π r 3
Write formula.
3 3
2
Substitute.
= —π (1)3
27
8
Evaluate the power.
= —π (1)
Multiply.
= —π
= —π —
=—
Inner Sphere
—
4
3
4
3
4
3
9
4
19
So, the volume of the inflated space is —π − —π = —π,
3
6
2
or about 10 cubic meters.
7. WHAT IF? The diameter of the inner sphere is 1.8 meters.
What is the volume of the inflated space?
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10.1 Exercises
Help with Homework
1. NUMBER SENSE Describe the difference between −34 and (−3)4.
2. WHICH ONE DOESN’T BELONG? Which one does not belong with the other
three? Explain your reasoning.
53
The base is 5.
53
The power is 5.
53
The exponent is 3.
53
Five is used as a
factor 3 times.
6)=3
9+(- 3)=
3+(- 9)=
4+(- =
1)
9+(-
Write the product using exponents.
1
⋅ ⋅ ⋅
( −—12 ) ⋅ ( −—12 ) ⋅ ( −—12 )
π ⋅π ⋅π ⋅x ⋅x ⋅x ⋅x
6.4 ⋅ 6.4 ⋅ 6.4 ⋅ 6.4 ⋅ b ⋅ b ⋅ b
⋅
⋅ ⋅
(−4) ⋅ (−4) ⋅ (−4) ⋅ y ⋅ y
(−t) ⋅ (−t) ⋅ (−t) ⋅ (−t) ⋅ (−t)
3. 3 3 3 3
4. (−6) (−6)
5.
1 1 1
6. — — —
3 3 3
7.
9.
8.
10.
Evaluate the expression.
2 11. 52
14.
12. −113
()
1 6
2
—
15.
13. (−1)6
()
( )
1 3
9
1 2
12
−—
17. ERROR ANALYSIS Describe and correct the
error in evaluating the expression.
16. − —
✗
⋅
−62 = (−6) (−6) = 36
18. PRIME FACTORIZATION Write the prime factorization of 675 using exponents.
(
1
4
⋅ 14 ⋅ 14 ⋅ 14 )
19. STRUCTURE Write − — — — — using exponents.
20. RUSSIAN DOLLS The largest doll is 12 inches
7
10
tall. The height of each of the other dolls is —
the height of the next larger doll. Write an
expression involving a power for the height
of the smallest doll. What is the height of the
smallest doll?
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Evaluate the expression.
⋅
3 21. 5 + 3 23
1
2
⋅
23. ( 132 − 122 ) ÷ 5
22. 2 + 7 (−3)2
⋅
24. — ( 43 − 6 32 )
25.
∣
1
2
— (7 + 5
3
)
∣
26.
∣ (− ) ÷ ( ) ∣
1 3
2
1 2
4
—
—
27. MONEY You have a part-time job. One day your boss offers to pay you either
2 h − 1 or 2h − 1 dollars for each hour h you work that day. Copy and complete
the table. Which option should you choose? Explain.
h
1
2
3
4
5
2h − 1
2h − 1
28. CARBON-14 DATING Scientists use carbon-14 dating to determine the age of
a sample of organic material.
a. The amount C (in grams) of a 100-gram sample of carbon-14 remaining
after t years is represented by the equation C = 100(0.99988)t. Use a
calculator to find the amount of carbon-14 remaining after 4 years.
b. What percent of the carbon-14 remains after 4 years?
29.
The frequency (in
vibrations per second) of a note on a
piano is represented by the equation
F = 440(1.0595)n, where n is the number
of notes above A-440. Each black or white
key represents one note.
a. How many notes do you take to
travel from A-440 to A?
A#
A
440
C#
B
C
D#
D
F#
E
F
G#
G
A
b. What is the frequency of A?
c. Describe the relationship between the number of notes
between A-440 and A and the increase in frequency.
Tell which property is illustrated by the statement. (Skills Review Handbook)
⋅
⋅
30. 8 x = x 8
⋅
⋅
⋅
31. (2 10)x = 2(10 x)
32. 3(x 1) = 3x
33. MULTIPLE CHOICE The polygons are similar. What is the value of x?
(Section 2.5)
A 15
○
B 16
○
C 17
○
D 36
○
27
24
18
x
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