Download section 7.2 Rational Exponents

480
Chapter 7
Radicals and Complex Numbers
section 7.2
Objectives
1n
mn
1. Definition of a and a
2. Converting Between
Rational Exponents and
Radical Notation
3. Properties of Rational
Exponents
4. Applications Involving
Rational Exponents
Rational Exponents
1. Definition of a1n and a mn
In Section 1.8, 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 a1n.
Definition of a1n
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
a1n 2a
n
Evaluating Expressions of the Form a1n
example 1
Skill Practice
Convert the expression to
radical form and simplify, if
possible.
1. 1614
Convert the expression to radical form and simplify, if possible.
a. 182 1 3
b. 811 4
c. 1001 2
d. 11002 1 2
e. 161 2
Solution:
2. 182 1 3
3
a. 182 13 1
8 2
3. 3612
4. 1492 12
4
b. 8114 1
81 3
5. 4912
c. 10012 1 10012
The exponent applies only to the base of 100.
12100
10
d. 11002 12 is not a real number because 1100 is not a real number.
e. 1612 1
1612
1
116
1
4
Write the expression with a positive exponent.
1
Recall that bn n .
b
n
If 1
a is a real number, then we can define an expression of the form amn in
such a way that the multiplication property of exponents still holds true. For
example,
1634
4
4
1163 2 14 2
163 2
4096 8
Answers
1. 2
2. 2
1
3.
6
4. Not a real number
4
11614 2 3 1 1
162 3 122 3 8
5. 7
Section 7.2
Rational Exponents
Definition of 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
amn 1a1n 2 m 1 1a2 m
n
and
amn 1am 2 1n 1am
n
The rational exponent in the expression amn is essentially performing two
operations. The numerator of the exponent raises the base to the mth power.
The denominator takes the nth root.
Evaluating Expressions of the Form a m/n
example 2
Convert each expression to radical form and simplify.
a. 823
b. 10052
c. a
1
b
25
32
d. 432
Skill Practice
Convert each expression to
radical form and simplify.
6. 932
7. 853
8. 3245
Solution:
3
a. 823 1 1
82 2
122 2
Take the cube root of 8 and square the result.
9. a
1 4 3
b
27
Simplify.
4
b. 10052 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 32
1 3
b
b a
A 25
25
1 3
a b
5
Take the square root of
1
25
and cube the result.
Simplify.
1
125
1
432
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
d. 432 Answers
6. 27
1
8.
16
7. 32
1
9.
81
481
482
Chapter 7
Radicals and Complex Numbers
Calculator Connections
A calculator can be used to confirm the results of
Example 2(a)–2(c).
2. Converting Between Rational Exponents and Radical Notation
Skill Practice
Convert each expression to
radical notation. Assume all
variables represent positive
real numbers.
10. t 45
11. 12y 3 2 14
12. 10p12
13. q23
example 3
Using Radical Notation and Rational Exponents
Convert each expression to radical notation. Assume all variables represent
positive real numbers.
b. 15x2 2 13
a. a35
5 3
5
a. a35 2
a or Q 2
a R3
3
b. 15x2 2 13 2
5x2
d. z34 Convert to an equivalent
expression using rational
exponents. Assume all
variables represent positive
real numbers.
3 2
14. 2
x
15. 15y
16. 51y
d. z34
Solution:
4
c. 3y14 31
y
Skill Practice
c. 3y14
Note that the coefficient 3 is not raised to the 14 power.
1
1
4
z3/4
2z3
example 4
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 3
a. 2
b
b. 17a
c. 71a
Solution:
4 3
a. 2
b b34
b. 17a 17a2 12
c. 71a 7a1 2
3. Properties of Rational Exponents
Answers
5 4
10. 2
t
4
11. 2
2y 3
1
12. 10 2p
13. 3
2q2
14. x 2 3
15. 15y2 1 2
1 2
16. 5y
In Section 1.8, several properties and definitions were introduced to simplify
expressions with integer exponents. These properties also apply to rational
exponents.
Section 7.2
Rational Exponents
483
Properties of Exponents and Definitions
Let a and b be nonzero real numbers. Let m and n be rational numbers such
that am, an, and bn are real numbers.
Description
Property
1. Multiplying like bases
mn
x x
1am 2 n amn
3. The power rule
1ab2 m ambm
4. Power of a product
5. Power of a quotient
a m
am
a b m
b
b
Description
Definition
m
1. Negative exponents
1
1
am a b m
a
a
2. Zero exponent
a0 1
x53
x35
x25
x15
am
amn
an
2. Dividing like bases
example 5
Example
13 43
a a a
m n
1213 2 12 216
a
1xy2 12 x12y12
4 12
412
2
b 12 25
5
25
Example
1 13 1
182 13 a b 8
2
50 1
Simplifying Expressions with Rational Exponents
Use the properties of exponents to simplify the expressions. Assume all
variables represent positive real numbers.
b. 1s4t8 2 14
a. y25y35
c. a
81cd2 13
b
3c2d4
a. y y
Use the properties of exponents
to simplify the expressions.
Assume all variables represent
positive real numbers.
17. x 1 2 x 34
18. 1a3b2 2 1 3
Solution:
25 35
Skill Practice
y
1252 1352
y55
Multiply like bases by adding
exponents.
19. a
32y 2 z3 14
b
2y2 z 5
Simplify.
y
b. 1s4t 8 2 14 s44t84
Apply the power rule. Multiply
exponents.
st 2
c. a
Simplify.
81cd2 13
b 127c1122d24 2 13
3c2d4
Simplify inside parentheses. Subtract
exponents.
127c3d 6 2 13
a
27c3 13
b
d6
2713c33
d63
3c
2
d
Simplify the negative exponent.
Apply the power rule. Multiply
exponents.
Simplify.
Answers
17. x 5/4
18. ab2/3
2y
19. 2
z
Chapter 7
484
Radicals and Complex Numbers
4. Applications Involving Rational Exponents
example 6
Skill Practice
20. The formula for the radius
of a sphere is
ra
3V 1 3
b
4p
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 1t
ra b 1
P
where V is the volume.
Find the radius of a sphere
whose volume is 113.04 in.3
(Use 3.14 for p.)
Find the annual rate of return on $5000 which grew to $12,500 after 6 years.
Solution:
A 1t
ra b 1
P
a
12,500 16
b 1
5000
where A $12,500, P $5000, and t 6
12.52 16 1
1.165 1
0.165 or 16.5%
The annual rate of return is 16.5%.
Calculator Connections
The expression
ra
12,500 16
b 1
5000
is easily evaluated on a graphing calculator.
Answer
20. 3 in.
section 7.2
Practice Exercises
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Study Skills Exercises
1. Don’t be afraid to mark up a book page. Use a highlighter to emphasize key points, definitions, rules,
formulas, or processes. What would you highlight in this section?
2. Define the key terms.
a. a1/n
b. am/n
Section 7.2
Rational Exponents
Review Exercises
For the exercises in this set, assume that all variables represent positive real numbers unless otherwise stated.
3
3. Given: 127
4. Given: 118
a. Identify the index.
a. Identify the index.
b. Identify the radicand.
b. Identify the radicand.
For Exercises 5–8, evaluate the radicals (if possible).
3
6. 18
5. 125
4
8. 1 1162 3
4
7. 181
Objective 1: Definition of a1/n and am/n
For Exercises 9–18, convert the expressions to radical form and simplify. (See Example 1.)
9. 1441 2
10. 1614
11. 1441 2
12. 1614
13. 11442 1 2
14. 1162 14
15. 1642 1 3
16. 1322 15
17. 1252 1 2
18. 1272 1 3
19. Explain how to interpret the expression amn as a radical.
3
3
20. Explain why 1 182 4 is easier to evaluate than 284.
For Exercises 21–24, simplify the expression, if possible. (See Example 2.)
21. a. 1634
b. 1634
c. 1162 34
d. 1634
e. 1634
f. 1162 34
22. a. 8134
b. 8134
c. 1812 34
d. 8134
e. 8134
f. 1812 34
23. a. 253 2
b. 253 2
c. 1252 3 2
d. 253 2
e. 253 2
f. 1252 3 2
24. a. 43 2
b. 43 2
c. 142 3 2
d. 43 2
e. 43 2
f. 142 3 2
For Exercises 25–50, simplify the expression. (See Example 2.)
25. 643 2
26. 813 2
27. 24335
29. 2743
30. 1654
31. a
33. 142 3 2
34. 1492 3 2
35. 182 13
100 3 2
b
9
28. 15 3
32. a
49 1 2
b
100
36. 192 12
485
486
Chapter 7
Radicals and Complex Numbers
37. 813
41.
1
3612
38. 912
42.
1
1612
1 23
1 12
45. a b a b
8
4
1 23
1 12
46. a b
a b
8
4
1 12
1 13
49. a b a b
4
64
50. a
39. 2723
43.
1
100013
47. a
1 34
1 1 2
b
a b
16
49
40. 12513
44.
1
8134
48. a
1 14
1 1 2
b a b
16
49
1 12
1 56
b a b
36
64
Objective 2: Converting Between Rational Exponents and Radical Notation
For Exercises 51–58, convert each expression to radical notation. (See Example 3.)
51. q2 3
52. t 35
53. 6y34
54. 8b4 9
55. 1x2y2 13
56. 1c2d2 16
57. 1qr2 1 5
58. 17x2 1 4
For Exercises 59–66, write each expression by using rational exponents rather than radical notation.
(See Example 4.)
3
59. 1
x
4
60. 1a
61. 101b
3
62. 21t
3 2
63. 2
y
6
64. 2z5
4
65. 2a2b3
66. 1abc
Objective 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.)
p53
q54
67. x14x54
68. 223 253
69.
71. 1y15 2 10
72. 1x12 2 8
73. 615635
74. a13a23
77. 1a13a14 2 12
78. 1x23x12 2 6
75.
4t13
t43
79. 15a2c12d1 2 2 2
76.
5s13
s53
80. 12x13y2z53 2 3
23
p
81. a
x23 12
b
y34
70.
q14
82. a
m14 4
b
n12
Section 7.2
83. a
16w2z 13
b
2wz8
87. 1x2y1 3 2 6 1x1 2yz2 3 2 2
84. a
50p1q 12
b
2pq3
85. 125x2y4z6 2 12
88. 1a1 3b1 2 2 4 1a12b35 2 10
89. a
x3my2m 1m
b
z5m
Rational Exponents
487
86. 18a6b3c9 2 23
90. a
a4nb3n 1n
b
cn
Objective 4: Applications Involving Rational Exponents
91. If the area A of a square is known, then the length of its sides, s, can be computed by the formula s A12.
a. Compute the length of the sides of a square having an area of 100 in.2
(See Example 6.)
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.
92. The radius r of a sphere of volume V is given by r a
3V 13
b . Find the radius of a sphere having a volume
4p
of 85 in.3 Round your answer to the nearest 0.1 in.
93. 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.
P
a. In one account, $10,000 grows to $16,802 after 5 years. 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 years. Compute the interest rate. Round your answer
to a tenth of a percent.
c. Which account produced a higher average yearly return?
94. Is 1a b2 12 the same as a12 b12? Why or why not?
Expanding Your Skills
For Exercises 95–100, write the expression as a single radical.
3
95. 2 1
x
3
96. 2
1x
5 3
99. 2
1w
3 4
100. 2
1w
4
97. 2
1y
4
98. 2 1
y
For Exercises 101–108, use a calculator to approximate the expressions and round to 4 decimal places, if necessary.
101. 912
102. 12513
103. 5014
104. 11722 35
3 2
105. 2
5
4 3
106. 2
6
107. 2103
3
108. 2
16