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Section 10.2
703
Rational Exponents
Rational Exponents
Section 10.2
1. Definition of a1/n and am/n
Concepts
In Sections 5.1–5.3, the properties for simplifying expressions with integer exponents
were presented. In this section, the properties are expanded to include expressions
with rational exponents. We begin by defining expressions of the form a1Ⲑn.
DEFINITION a1/n
n
Let a be a real number, and let n be an integer such that n 7 1. If 1a is a
real number, then
1. Definition of a1/n and am /n
2. Converting Between
Rational Exponents and
Radical Notation
3. Properties of Rational
Exponents
4. Applications Involving
Rational Exponents
a1Ⲑn ⫽ 2a
n
Evaluating Expressions of the Form a1/n
Example 1
Convert the expression to radical form and simplify, if possible.
a. 1⫺82 1Ⲑ 3
b. 811Ⲑ 4
c. ⫺1001 Ⲑ 2
d. 1⫺1002 1Ⲑ 2
e. 16⫺1Ⲑ 2
Solution:
3
a. 1⫺82 1Ⲑ3 ⫽ 1⫺8 ⫽ ⫺2
4
b. 811Ⲑ4 ⫽ 181 ⫽ 3
c. ⫺1001Ⲑ2 ⫽ ⫺1 ⴢ 1001Ⲑ2
The exponent applies only to the base of 100.
⫽ ⫺12100
d. 1⫺1002
⫽ ⫺10
1Ⲑ2
is not a real number because 1⫺100 is not a real number.
e. 16⫺1Ⲑ2 ⫽
1
161Ⲑ2
⫽
1
116
⫽
1
4
Write the expression with a positive exponent.
1
Recall that b⫺n ⫽ n .
b
Skill Practice Convert the expression to radical form and simplify, if possible.
1. 1⫺642 1Ⲑ 3
2. 161Ⲑ4
3. ⫺361Ⲑ2
4. 1⫺362 1Ⲑ2
5. 64⫺1Ⲑ3
n
a is a real number, then we can define an expression of the form amⲐn in such a
If 1
way that the multiplication property of exponents still holds true. For example:
163Ⲑ4
4
1161Ⲑ4 2 3 ⫽ 1 1
162 3 ⫽ 122 3 ⫽ 8
4
4
1163 2 1Ⲑ4 ⫽ 2163 ⫽ 24096 ⫽ 8
Answers
1. ⫺4
2. 2
4. Not a real number
3. ⫺6
5.
1
4
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Chapter 10 Radicals and Complex Numbers
DEFINITION a m/n
Let a be a real number, and let m and n be positive integers such that m and
n
n share no common factors and n 7 1. If 1
a is a real number, then
amⲐn ⫽ 1a1Ⲑn 2 m ⫽ 1 1a2 m
n
and
amⲐn ⫽ 1am 2 1Ⲑn ⫽ 1am
n
The rational exponent in the expression amⲐn is essentially performing two
operations. The numerator of the exponent raises the base to the mth power. The
denominator takes the nth root.
Example 2
Evaluating Expressions of the Form a m/n
Convert each expression to radical form and simplify.
a. 82Ⲑ3
b. 1005Ⲑ2
c. a
1 3Ⲑ2
b
25
d. 4⫺3Ⲑ2
e. 1⫺812 3/4
Solution:
Calculator Connections
A calculator can be used
to confirm the results of
Example 2(a)–2(c).
3
a. 82Ⲑ3 ⫽ 1 1
82 2
⫽ 122 2
Take the cube root of 8 and square the result.
Simplify.
⫽4
b. 1005Ⲑ2 ⫽ 1 11002 5
⫽ 1102 5
Take the square root of 100 and raise the result to
the fifth power.
Simplify.
⫽ 100,000
c. a
1 3Ⲑ2
1 3
b ⫽a
b
25
A 25
1 3
⫽a b
5
⫽
Take the square root of
1
and cube the result.
25
Simplify.
1
125
1 3Ⲑ2
1
d. 4⫺3Ⲑ2 ⫽ a b ⫽ 3 2
4
4Ⲑ
Write the expression with positive exponents.
⫽
1
1 142 3
Take the square root of 4 and cube the result.
⫽
1
23
Simplify.
⫽
1
8
4
e. 1⫺812 3/4 is not a real number because 1
⫺81 is not a real number.
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Rational Exponents
Skill Practice Convert each expression to radical form and simplify.
6. 93Ⲑ2
8. a
7. 85Ⲑ3
1 4Ⲑ 3
b
27
9. 32⫺4Ⲑ5
10. 1⫺42 3/2
2. Converting Between Rational Exponents
and Radical Notation
Example 3
Using Radical Notation and Rational Exponents
Convert each expression to radical notation. Assume all variables represent positive real numbers.
b. 15x2 2 1Ⲑ3
a. a3Ⲑ5
c. 3y1Ⲑ4
d. z⫺3Ⲑ4
Solution:
5
5
a. a3Ⲑ5 ⫽ 2a3 or Q 2a R 3
3
b. 15x2 2 1Ⲑ3 ⫽ 25x2
4
c. 3y1Ⲑ4 ⫽ 31y
d. z⫺3Ⲑ4 ⫽
Note that the coefficient 3 is not raised to the 14 power.
1
1
⫽ 4
3/4
z
2z3
Skill Practice Convert each expression to radical notation. Assume all
variables represent positive real numbers.
11. t 4Ⲑ5
12. 12y 3 2 1Ⲑ4
Example 4
13. 10p1Ⲑ2
14. q⫺2Ⲑ3
Using Radical Notation and Rational Exponents
Convert each expression to an equivalent expression by using rational
exponents. Assume that all variables represent positive real numbers.
4
a. 2b3
Solution:
4
a. 2b3 ⫽ b3Ⲑ4
b. 17a
c. 71a
b. 17a ⫽ 17a2 1Ⲑ2
c. 71a ⫽ 7a1Ⲑ 2
Skill Practice Convert to an equivalent expression using rational exponents.
Assume all variables represent positive real numbers.
3
15. 2 x2
16. 15y
17. 51y
3. Properties of Rational Exponents
In Sections 5.1–5.3, several properties and definitions were introduced to simplify
expressions with integer exponents.These properties also apply to rational exponents.
Answers
6. 27
9.
1
16
5
11. 2t 4
14.
1
3
2q 2
17. 5y 1Ⲑ 2
7. 32
8.
1
81
10. Not a real number
4
12. 2
2y 3
13. 102p
15. x 2 Ⲑ 3
16. 15y2 1Ⲑ 2
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Chapter 10 Radicals and Complex Numbers
SUMMARY
Definitions and Properties of Exponents
Let a and b be nonzero real numbers. Let m and n be rational numbers such
that am, an, and bm are real numbers.
Description
Property
1. Multiplying like bases
2. Dividing like bases
m⫹n
4. Power of a product
5. Power of a quotient
Description
⫽ x5Ⲑ3
x3Ⲑ5
⫽ x2 Ⲑ 5
x1Ⲑ5
121Ⲑ3 2 1Ⲑ2 ⫽ 21Ⲑ6
1ab2 m ⫽ ambm
19y2 1/2 ⫽ 91/2y1/2 ⫽ 3y1/2
Definition
Example
a m
am
a b ⫽ m
b
b
a
4 1Ⲑ2
41 Ⲑ 2
2
b ⫽ 1Ⲑ2 ⫽
25
5
25
1
1
a⫺m ⫽ a b ⫽ m
a
a
1 1Ⲑ3 1
182 ⫺1Ⲑ3 ⫽ a b ⫽
8
2
a0 ⫽ 1
50 ⫽ 1
m
1. Negative exponents
x x
am
⫽ am⫺n
an
1am 2 n ⫽ amn
3. The power rule
Example
1Ⲑ3 4Ⲑ3
a a ⫽a
m n
2. Zero exponent
Simplifying Expressions with Rational Exponents
Example 5
Use the properties of exponents to simplify the expressions. Assume all variables
represent positive real numbers.
a. y2Ⲑ5y3Ⲑ5
b.
6a⫺1/2
a3/2
c. a
s1/2t1/3 4
b
w3/4
Solution:
a. y2Ⲑ5y3Ⲑ5 ⫽ y12Ⲑ52 ⫹ 13Ⲑ52
Multiply like bases by adding exponents.
⫽ y5Ⲑ5
Simplify.
⫽y
b.
6a⫺1/2
a3/2
Divide like bases by subracting exponents.
⫽ 6a1⫺1/22 ⫺ 13/22
4
Simplify: ⫺ ⫽ ⫺2
2
⫽ 6a⫺2
⫽
c. a
6
a2
Simplify the negative exponent.
s1/2 t 1/3 4 s11/22 ⴢ4t 11/32 ⴢ4
b ⫽
w3/4
w13/42 ⴢ4
⫽
s2t 4/3
w3
Apply the power rule. Multiply exponents.
Simplify.
Skill Practice Use the properties of exponents to simplify the expressions.
Assume all variables represent positive real numbers.
Answers
18. x 5/4
19.
4
k
2 3
20.
ab
c15/4
18. x 1Ⲑ 2 ⴢ x 3Ⲑ4
19.
4k⫺2/3
k1/3
20. a
a1/3b1/2 6
b
c 5/8
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4. Applications Involving Rational Exponents
Example 6
Applying Rational Exponents
Suppose P dollars in principal is invested in an account that earns interest
annually. If after t years the investment grows to A dollars, then the annual rate
of return r on the investment is given by
A 1Ⲑt
r⫽a b ⫺1
P
Find the annual rate of return on $5000 which grew to $6894.21 after 6 yr.
Solution:
A 1Ⲑt
r⫽a b ⫺1
P
⫽a
6894.21 1Ⲑ6
b ⫺1
5000
where A ⫽ $6894.21, P ⫽ $5000, and t ⫽ 6
⬇ 0.055 or 5.5%
The annual rate of return is 5.5%.
Skill Practice
21. The formula for the radius of a sphere is
r⫽a
3V 1Ⲑ 3
b
4p
where V is the volume. Find the radius of a sphere whose volume is 113.04 in.3
(Use 3.14 for p.)
Answer
21. 3 in.
Section 10.2
Practice Exercises
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For the exercises in this set, assume that all variables represent positive real numbers unless otherwise stated.
Study Skills Exercises
1. Before you do your homework for this section, go back to Sections 5.1–5.3 and review the properties of exponents.
Do several problems from the Section 5.3 exercises. This will help you with the concepts in Section 10.2.
2. Define the key terms.
a. a1 /n
b. a m/n
Review Exercises
3
3. Given: 1
27
4. Given: 118
a. Identify the index.
a. Identify the index.
b. Identify the radicand.
b. Identify the radicand.
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Chapter 10 Radicals and Complex Numbers
For Exercises 5–8, evaluate the radicals.
5. 125
3
6. 18
4
7. 181
4
8. 1 1162 3
Concept 1: Definition of a1/n and a m/n
For Exercises 9–20, convert the expressions to radical form and simplify. (See Example 1.)
9. 1441Ⲑ 2
10. 161Ⲑ4
11. ⫺1441Ⲑ 2
12. ⫺161Ⲑ4
13. 1⫺1442 1Ⲑ 2
14. 1⫺162 1Ⲑ4
15. 1⫺642 1Ⲑ 3
16. 1⫺322 1Ⲑ5
17. 1252 ⫺1Ⲑ 2
18. 1272 ⫺1Ⲑ 3
19. ⫺49⫺1/2
20. ⫺64⫺1/2
21. Explain how to interpret the expression amⲐn as a radical.
3
3
22. Explain why 1 182 4 is easier to evaluate than 284.
For Exercises 23–26, simplify the expression, if possible. (See Example 2.)
23. a. 163Ⲑ4
24. a. 813Ⲑ4
b. ⫺163Ⲑ4
25. a. 253Ⲑ 2
b. ⫺813Ⲑ4
26. a. 43Ⲑ 2
b. ⫺253Ⲑ 2
b. ⫺43Ⲑ 2
c. 1⫺162 3Ⲑ4
c. 1⫺812 3Ⲑ4
c. 1⫺252 3Ⲑ 2
c. 1⫺42 3Ⲑ 2
e. ⫺16⫺3Ⲑ4
e. ⫺81⫺3Ⲑ4
e. ⫺25⫺3Ⲑ 2
e. ⫺4⫺3Ⲑ 2
d. 16⫺3Ⲑ4
d. 81⫺3Ⲑ4
f. 1⫺162 ⫺3Ⲑ4
d. 25⫺3Ⲑ 2
f. 1⫺812 ⫺3Ⲑ4
d. 4⫺3Ⲑ 2
f. 1⫺252 ⫺3Ⲑ 2
f. 1⫺42 ⫺3Ⲑ 2
For Exercises 27–50, simplify the expression. (See Example 2.)
27. 64⫺3Ⲑ 2
28. 81⫺3Ⲑ 2
29. 2433Ⲑ5
31. ⫺27⫺4Ⲑ3
32. ⫺16⫺5Ⲑ4
33. a
35. 1⫺42 ⫺3Ⲑ 2
36. 1⫺492 ⫺3Ⲑ 2
37. 1⫺82 1Ⲑ3
39. ⫺81Ⲑ3
40. ⫺91Ⲑ2
41.
43.
1
1000⫺1Ⲑ3
47. a
1 ⫺3Ⲑ4
1 ⫺1Ⲑ 2
b
⫺a b
16
49
44.
1
81⫺3Ⲑ4
48. a
1 1Ⲑ4
1 1Ⲑ 2
b ⫺a b
16
49
100 ⫺3Ⲑ 2
b
9
1
36⫺1Ⲑ2
30. 15Ⲑ 3
34. a
49 ⫺1Ⲑ 2
b
100
38. 1⫺92 1Ⲑ2
42.
1
16⫺1Ⲑ2
1 2Ⲑ3
1 1Ⲑ2
45. a b ⫹ a b
8
4
1 ⫺2Ⲑ3
1 ⫺1Ⲑ2
⫹a b
46. a b
8
4
1 1Ⲑ2
1 ⫺1Ⲑ3
49. a b ⫹ a b
4
64
50. a
1 1Ⲑ2
1 ⫺5Ⲑ6
b ⫹a b
36
64
Concept 2: Converting Between Rational Exponents and Radical Notation
For Exercises 51–58, convert each expression to radical notation. (See Example 3.)
51. q2Ⲑ 3
52. t 3Ⲑ5
53. 6y3Ⲑ4
54. 8b4 Ⲑ 9
55. 1x 2y2 1Ⲑ3
56. 1c 2d2 1Ⲑ6
57. 1qr2 ⫺1 Ⲑ5
58. 17x2 ⫺1 Ⲑ4
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For Exercises 59–66, write each expression by using rational exponents rather than radical notation.
(See Example 4.)
Section 10.2
3
4
3
59. 1x
60. 1a
61. 101b
62. ⫺21t
3
63. 2y2
6
64. 2z5
4
65. 2a2b3
66. 1abc
Concept 3: Properties of Rational Exponents
For Exercises 67–90, simplify the expressions by using the properties of rational exponents. Write the final answer
using positive exponents only. (See Example 5.)
p5Ⲑ3
q5Ⲑ4
67. x1Ⲑ4x⫺5Ⲑ4
68. 22Ⲑ3 2⫺5Ⲑ3
69.
71. 1y1Ⲑ5 2 10
72. 1x1Ⲑ2 2 8
73. 6⫺1Ⲑ563Ⲑ5
74. a⫺1Ⲑ3a2Ⲑ3
77. 1a1Ⲑ3a1Ⲑ4 2 12
78. 1x2Ⲑ3x1Ⲑ2 2 6
75.
4t⫺1Ⲑ3
t4Ⲑ3
79. 15a2c⫺1Ⲑ2d1Ⲑ 2 2 2
83. a
16w⫺2z 1Ⲑ3
b
2wz⫺8
87. 1x2y⫺1Ⲑ 3 2 6 1x1Ⲑ 2yz2Ⲑ 3 2 2
76.
5s⫺1Ⲑ3
s5Ⲑ3
80. 12x⫺1Ⲑ3y2z5Ⲑ3 2 3
84. a
50p⫺1q 1Ⲑ2
b
2pq⫺3
88. 1a⫺1Ⲑ 3b1Ⲑ 2 2 4 1a⫺1Ⲑ2b3Ⲑ5 2 10
p2Ⲑ3
81. a
x⫺2Ⲑ3 12
b
y⫺3Ⲑ4
85. 125x2y4z6 2 1Ⲑ2
89. a
x3my2m 1Ⲑm
b
z5m
70.
q1Ⲑ4
82. a
m⫺1Ⲑ4 ⫺4
b
n⫺1Ⲑ2
86. 18a6b3c9 2 2Ⲑ3
90. a
a4nb3n 1Ⲑn
b
cn
Concept 4: Applications Involving Rational Exponents
91. If P dollars in principal grows to A dollars after t years with annual interest, then the interest rate is given
A 1/t
by r ⫽ a b ⫺ 1. (See Example 6.)
P
a. In one account, $10,000 grows to $16,802 after 5 yr. Compute the interest rate. Round your answer to a
tenth of a percent.
b. In another account $10,000 grows to $18,000 after 7 yr. Compute the interest rate. Round your answer
to a tenth of a percent.
c. Which account produced a higher average yearly return?
92. If the area A of a square is known, then the length of its sides, s, can be computed by the formula s ⫽ A1Ⲑ2.
a. Compute the length of the sides of a square having an area of 100 in.2
b. Compute the length of the sides of a square having an area of 72 in.2 Round your answer to the nearest
0.1 in.
93. The radius r of a sphere of volume V is given by r ⫽ a
3V 1Ⲑ3
b . Find the radius of a sphere having a volume
4p
of 85 in.3 Round your answer to the nearest 0.1 in.
94. Is 1a ⫹ b2 1Ⲑ2 the same as a1Ⲑ2 ⫹ b1Ⲑ2? If not, give a counterexample.
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Chapter 10 Radicals and Complex Numbers
Expanding Your Skills
For Exercises 95–106, write the expression using rational exponents. Then simplify and convert back to radical
notation. Assume that all variables represent positive real numbers.
15
Example: 2x10
Rational
exponents
6
95. 2 y3
9
99. 2 x6
103. 216x8y6
x10/15
Simplify
3 2
2
x
12
4
96. 2w2
12
Radical
notation
x2/3
18
97. 2z3
98. 2t 3
6
102. 2m2p8
3
106. 264m5n9p
100. 2p9
101. 2x3y6
104. 281a12b20
105. 2 8x3y7z8
8
3
For Exercises 107–110, write the expression as a single radical.
3
107. 21x
3
108. 2 1x
5 3
109. 2 1w
3 4
110. 2 1w
For Exercises 111–118, use a calculator to approximate the expressions. Round to four decimal places, if necessary.
111. 91Ⲑ2
112. 125⫺1Ⲑ3
113. 50⫺1Ⲑ4
114. 11722 3Ⲑ5
3 2
115. 2
5
4 3
116. 2
6
117. 2103
3
118. 2
16
Section 10.3
Simplifying Radical Expressions
Concepts
1. Multiplication Property of Radicals
1. Multiplication Property of
Radicals
2. Simplifying Radicals by
Using the Multiplication
Property of Radicals
3. Simplifying Radicals by
Using the Order of
Operations
You may have already noticed certain properties of radicals involving a product or
quotient.
PROPERTY Multiplication Property of Radicals
n
n
Let a and b represent real numbers such that 1a and 1b are both real. Then
n
n
n
1
ab ⫽ 1
aⴢ 1
b
The multiplication property of radicals follows from the property of rational
exponents.
1ab ⫽ 1ab2 1Ⲑn ⫽ a1Ⲑnb1Ⲑn ⫽ 1a ⴢ 1b
n
n
n
The multiplication property of radicals indicates that a product within a radicand
can be written as a product of radicals, provided the roots are real numbers. For
example:
1144 ⫽ 116 ⴢ 19
The reverse process is also true. A product of radicals can be written as a single
radical provided the roots are real numbers and they have the same indices.
13 ⴢ 112 ⫽ 136