LESSON
4.6
Negative and Zero Exponents
Now
BEFORE
Review Vocabulary
power, p. 10
exponent, p. 10
base, p. 10
WHY?
You worked with positive You’ll work with negative and
exponents.
zero exponents.
So you can compare nanoseconds
to milliseconds, as in Ex. 15.
Consider the following pattern of powers of 2.
23 ⴝ 8
22 ⴝ 4
21 ⴝ 2
20 ⴝ ?
2ⴚ1 ⴝ ?
2ⴚ2 ⴝ ?
As exponents
decrease by 1 . . .
the values of the
powers are halved.
1
2
By extending the pattern, you can conclude that 20 1, 21 , and
1
1
1
1
1
22 . Because _1 and _2 , the pattern suggests the following
4
2
2
4
2
definitions for negative and zero exponents.
Negative and Zero Exponents
For any nonzero number a, a0 1.
1
For any nonzero number a and any integer n, an _n .
a
Example 1
Reading Algebra
Powers with Negative and Zero Exponents
Write the expression using only positive exponents.
1
a. 35 _5
In an expression like 16x6y,
the exponent is associated
only with the variable that
comes directly before it, not
with the coefficient of the
variable.
Definition of negative exponent
3
b. m0n4 1 p n4
1
_4
c. 16x6y n
16y
_
x6
Definition of zero exponent
Definition of negative exponent
Definition of negative exponent
Checkpoint
Write the expression using only positive exponents.
1. 52
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2. 1,000,000 0
Lesson 4.6
3. 3y2
4. a7b 3
Negative and Zero Exponents
201
Rewriting Fractions You can use the prime factorization of a number to
write a fraction as an expression involving negative exponents.
Example 2
Rewriting Fractions
Write the expression without using a fraction bar.
2
1
a
b. _
3
a. 1
6
c
Solution
1
1
a. 1
_4
6
2
Write prime factorization of 16.
24
Definition of negative exponent
2
a
b. _3 a 2c3
Definition of negative exponent
c
Products and Quotients of Powers You can use the product of powers
property and the quotient of powers property to find products and
quotients that involve negative exponents.
Example 3
Using Powers Properties with Negative Exponents
Find the product or quotient. Write your answer using only positive
exponents.
3
0.8n
b. __
2
a. 5 10 p 5 6
n
Solution
a. 510 p 56 510 (6)
4
5
Product of powers property
Add exponents.
3
0.8n
b. __
0.8n3 2
2
n
Quotient of powers property
0.8n5
Subtract exponents.
0.8
_
5
Definition of negative exponent
n
Checkpoint
Write the expression without using a fraction bar.
1
5. 25
1
6. _
1000
7
x
8. _
2
2
7. _
8
a
z
Find the product or quotient. Write your answer using only positive
exponents.
9. 37 p 311
3
(0.2)
13. __
4
(0.2)
202
Chapter 4
Factors, Fractions, and Exponents
10. (0.5)8 p (0.5)7 11. m3 p m1
2
7
14. _
8
7
3
5k
15. _
9
k
12. a2 p a 10
2
b
16. _
6
b
Example 4
Solving Problems Involving Negative Exponents
Geckos Geckos can easily climb smooth vertical surfaces. Biologists
have discovered that tiny hairs are the reason that the feet of a gecko
are so sticky. Each hair is about 100 micrometers long. A micrometer
is 106 meter. What is the length of one hair in meters?
Solution
To find the length of one hair in meters, multiply the length of the
hair in micrometers by the number of micrometers in one meter.
100 p 106 10 2 p 106
10 2 (6)
Rewrite 100 as 102.
Product of powers property
4
10
Add exponents.
1
_4
Definition of negative exponent
1
__
10,000
Evaluate power.
10
1
Answer The length of one hair is about __
meter.
10,000
4.6
Exercises
More Practice, p. 842
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Practice Exercises
Guided Practice
Vocabulary Check
1. Write 7 2 using a positive exponent.
2. If a is nonzero, does the value of a0 depend of the value of a? Explain.
Skill Check
Write the expression using only positive exponents.
3. 53
4. 35
5. 4a6
6. b3c 0
Write the expression without using a fraction bar.
1
7. 27
1
8. _
8
4
9. _
3
x
10
11
10. _
5
c
Find the product. Write your answer using only positive exponents.
11. 64 p 67
Guided Problem Solving
12. 32 p 38
13. x 11 p x 3
15. How many nanoseconds are in a millisecond?
1
2
14. z 5 p z 1
Name of unit
Duration
Write the quotient of the duration of a
millisecond and the duration of a
nanosecond.
Millisecond
103 sec
Microsecond
106 sec
Use the quotient of powers property to
simplify the quotient in Step 1.
Nanosecond
109 sec
Lesson 4.6
Negative and Zero Exponents
203
Practice and Problem Solving
Homework Help
Example
1
2
3
4
Exercises
16–23
24–31
32–39, 43–50
40, 55
Write the expression using only positive exponents.
16. 136
17. 1210
18. 89
19. 204
20. xy 0
21. 18f 1
22. 0.6g 5
23. c 3d 1
Write the expression without using a fraction bar.
1
24. 25
1
25. 19
4
29. d
8
28. _5
Lesson Resources
c
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• More Examples
• @HomeTutor
1
26. __
1
27. 64
30.
9a
31. _
6
10,000
4y
_
x
2
3
b
Find the product. Write your answer using only positive exponents.
32. 34 p 37
33. 5 p 55
34. 102 p 108
36. 2s5 p s 3
37. 5t 3 p 3t 8
38. 0.4a 0 p 0.7a4 39. b5 p b9
35. 130 p 136
40. Historical Documents Scientists have discovered that nanoparticles of
a substance called slaked lime can help preserve historical documents.
1
meter.
The diameters of these nanoparticles are less than ___
100,000,000,000
Write this number using a negative exponent.
41. Critical Thinking Explain how 6 2 is different from 62.
42.
Writing
1
Explain why the rule an _n does not apply to a 0.
a
Find the quotient. Write your answer using only positive exponents.
2
5
(0.4)
44. __
6
2
43. _
8
2
(0.4)
3
1.7a
47. _
7
a
5
15b
48. __
4
3b
9
3
16
45. _
8
15
46. _
4
15
16
4
26w
49. __
12
13w
2
11g
50. _
4
g
Use a calculator to evaluate the expression. If necessary, round the
result to the nearest thousandth.
51. (4.5)3
52. (8.1)2
53. (3.2)4
54. (7.5)3
55. Brittle Stars The brittle star is a type of starfish. A certain species
of brittle star has a skeleton that is covered in microscopic crystals.
Scientists have discovered that these crystals act as lenses that allow
the brittle star to sense light.
1
a. The surface of each crystal has an area of ___
square meter.
1,000,000,000
Write this number using a negative exponent.
In the
Real World
Brittle Stars There are
over 2000 species of brittle
stars. They live anywhere from
the shoreline to 6755 feet
underwater. How many miles
is this distance?
204
Chapter 4
b. Approximately 104 crystals cover the skeleton of a brittle star. What
is the total area of all the crystals on a single brittle star? Write your
answer using a negative exponent.
Find the quotient. Write your answer using only positive exponents.
6 4
a b
56. _
3 7
a b
Factors, Fractions, and Exponents
2 11
c d
57. _
8 5
c d
8 4
m n
58. _
2 9
m n
2
x y
59. _
10 7
x y
60. Extended Problem Solving Inkjet printers spray droplets of ink
onto paper. The volume of a single droplet is about 10 picoliters.
Some printers spray as many as 106 droplets to completely cover a
square inch of paper.
a. Find the volume of ink in picoliters needed to completely cover a
square inch of paper.
b. There are 1012 picoliters in a liter. Find the volume of ink in liters
needed to completely cover a square inch of paper.
c. Estimate Find the area of a 8.5 inch by 11 inch piece of paper and
round this number to the nearest power of 10. Then estimate the
volume of ink (in liters) needed to completely cover an entire piece
of paper.
This printer can print pages
several feet wide.
d. Apply Suppose an inkjet cartridge contains 60 milliliters of ink.
About how many pages can this cartridge print if each page is
8.5 inches by 11 inches and is covered completely in ink?
61. Challenge One way to develop the definitions of zero and negative
exponents is to use the quotient of powers property.
n
a
a. Consider _
n . First, simplify using the quotient of powers property.
a
b. Then simplify the expression in part (a) in a different way: Write
the numerator and denominator as a product of a’s and divide out
common factors. Which definition of exponents have you developed?
0
a
c. Now consider _
. First, simplify using the quotient of powers
an
property.
d. Then simplify the expression in part (c) by using the definition of a
zero exponent. Which definition of exponents have you developed?
Mixed Review
Algebra Basics Solve the equation using mental math. (Lesson 2.4)
62. 9 x 17
63. 8 x 3
64. 3x 36
x
65. 6
8
66. Amusement Parks You must be at least 46 inches tall to ride the
bumper cars at an amusement park. Write and graph an inequality to
show the heights for which you can ride the bumper cars. (Lesson 3.4)
Find the product or quotient. Write your answer using exponents.
(Lesson 4.5)
67. 32 p 32
Standardized Test
Practice
8
9
10
70. _5
2
69. _4
68. 54 p 5
2
10
71. Multiple Choice Which expression is not equivalent to x 2 p x 6?
A. x4
B. x2 p x 6
2
x
D. _6
1
C. _4
x
x
6
24a
72. Multiple Choice Which expression is equivalent to _
?
2
3b
6 2
F. 24a b
6 2
G. 24a b
6 2
H. 8a b
Lesson 4.6
I. 8a6b 2
Negative and Zero Exponents
205