478
(9–12)
Chapter 9
In this
Radicals and Rational Exponents
9.2
RATIONAL EXPONENTS
section
●
Rational Exponents
●
Using the Rules of Exponents
●
Simplifying Expressions
Involving Variables
You have learned how to use exponents to express powers of numbers and radicals
to express roots. In this section you will see that roots can be expressed with exponents also. The advantage of using exponents to express roots is that the rules of
exponents can be applied to the expressions.
Rational Exponents
The nth root of a number can be expressed by using radical notation or the exponent
3
8 both represent the cube root of 8, and we have
1n. For example, 813 and calculator
3
8 2.
813 close-up
You can find the fifth root of 2
using radical notation or exponent notation. Note that
the fractional exponent 15
must be in parentheses.
Definition of a1/ n
If n is any positive integer, then
n
a,
a1n n
a is a real number.
provided that Later in this section we will see that using exponent 1n for nth root is compatible with the rules for integral exponents that we already know.
E X A M P L E
1
Radicals or exponents
Write each radical expression using exponent notation and each exponential
expression using radical notation.
3
35
a) 4
b) xy
Solution
3
a) 3513
35
b) xy
(xy)14
4
c) 512
d) a15
c) 512 5
d) a15 a ■
5
In the next example we evaluate some exponential expressions.
E X A M P L E
2
Finding roots
Evaluate each expression.
b) (8)13
a) 412
c) 8114
d) (9)12
Solution
a) 412 4
2
3
13
2
b) (8) 8
4
3
c) 8114 81
is an even root of a negative number, it is not a real
d) Because (9)12 or 9
■
number.
9.2
Rational Exponents
(9–13)
479
We now extend the definition of exponent 1n to include any rational number
as an exponent. The numerator of the rational number indicates the power, and the
denominator indicates the root. For example, the expression
↓|
823
←
Power
Root
represents the square of the cube root of 8. So we have
823 (813)2 (2)2 4.
helpful
hint
Note that in amn we do not
require mn to be reduced. As
long as the nth root of a is real,
then the value of amn is the
same whether or not mn is in
lowest terms.
Definition of am/ n
If m and n are positive integers, then
amn (a1n)m ,
provided that a1n is a real number.
We define negative rational exponents just like negative integral exponents.
Definition of am/ n
If m and n are positive integers and a 0, then
1
amn m,
a n
provided that a1n is a real number.
E X A M P L E
3
Radicals or exponents
Write each radical expression using exponent notation and each exponential
expression using radical notation.
1
3
x2
b) 3
a) 4
m
c) 523
d) a25
Solution
1
1
b) 3 34
m34
4
m
m
1
d) a25 2
5
a
3
a) x2 x 23
3
c) 523 52
■
To evaluate an expression with a negative rational exponent, remember that the
denominator indicates root, the numerator indicates power, and the negative sign
indicates reciprocal:
amn
↑↑ ↑
Root
Power
Reciprocal
480
(9–14)
Chapter 9
Radicals and Rational Exponents
The root, power, and reciprocal can be evaluated in any order. However, to evaluate
amn mentally it is usually simplest to use the following strategy.
Strategy for Evaluating am/ n Mentally
1. Find the nth root of a.
2. Raise your result to the mth power.
3. Find the reciprocal.
For example, to evaluate 823 mentally, we find the cube root of 8 (which is 2),
1
square 2 to get 4, then find the reciprocal of 4 to get . In print 823 could be
4
1
written for evaluation as ((813)2)1 or 132.
(8 )
E X A M P L E
4
calculator
close-up
A negative fractional exponent indicates a reciprocal, a
root, and a power. To find 432
you can find the reciprocal
first, the square root first, or
the third power first as shown
here.
Rational exponents
Evaluate each expression.
b) 432
a) 2723
c) 8134
d) (8)53
Solution
a) Because the exponent is 23, we find the cube root of 27 and then square it:
2723 (2713)2 32 9
b) Because the exponent is 32, we find the square root of 4, cube it, and find the
reciprocal:
1
1
1
432 12
(4 )3 23 8
c) Because the exponent is 34, we find the fourth root of 81, cube it, and find the
reciprocal:
1
1
1
8134 14
3 3 3
27
(81 )
Definition of negative exponent
1
1
1
1
d) (8)53 13 5 5 ((8) )
(2)
32
32
■
CAUTION
An expression with a negative base and a negative exponent
can have a positive or a negative value. For example,
1
(8)53 32
and
1
(8)23 .
4
Using the Rules of Exponents
All of the rules for exponents hold for rational exponents as well as integral exponents. Of course, we cannot apply the rules of exponents to expressions that are not
real numbers.
9.2
Rational Exponents
(9–15)
481
Rules for Rational Exponents
The following rules hold for any nonzero real numbers a and b and rational
numbers r and s for which the expressions represent real numbers.
1. aras ar s
ar
2. s ars
a
3. (ar ) s ars
4. (ab)r arbr
a r ar
5. r
b
b
Product rule
Quotient rule
Power of a power rule
Power of a product rule
Power of a quotient rule
We can use the product rule to add rational exponents. For example,
1614 1614 1624.
The fourth root of 16 is 2, and 2 squared is 4. So 1624 4. Because we also have
1612 4, we see that a rational exponent can be reduced to its lowest terms. If an
exponent can be reduced, it is usually simpler to reduce the exponent before we
evaluate the expression. We can simplify 1614 1614 as follows:
1614 1614 1624 1612 4
E X A M P L E
5
Using the product and quotient rules with rational exponents
Simplify each expression.
534
b) 1
a) 2716 2712
5 4
Solution
a) 2716 2712 2716 12
2723
9
3 4
5
b) 1
53414 524 512 5
5 4
E X A M P L E
6
Product rule for exponents
We used the quotient rule to
subtract the exponents.
■
Using the power rules with rational exponents
Simplify each expression.
a) 312 1212
b) (310 )12
26
c) 9
3
13
Solution
a) Because the bases 3 and 12 are different, we cannot use the product rule to add
the exponents. Instead, we use the power of a product rule to place the 12
power outside the parentheses:
312 1212 (3 12)12 3612 6
482
(9–16)
Chapter 9
Radicals and Rational Exponents
b) Use the power of a power rule to multiply the exponents:
(310)12 35
26
c) 9
3
13
(26)13
(39)13
22
3
3
33
2
2
27
4
Power of a quotient rule
Power of a power rule
Definition of negative exponent
■
Simplifying Expressions Involving Variables
helpful
hint
We usually think of squaring
and taking a square root as
inverse operations, which they
are as long as we stick to positive numbers.We can square 3
to get 9, and then find the
square root of 9 to get 3—
what we started with. We
don’t get back to where we
began if we start with 3.
When simplifying expressions involving rational exponents and variables, we must
be careful to write equivalent expressions. For example, in the equation
(x2)12 x
it looks as if we are correctly applying the power of a power rule. However, this
statement is false if x is negative because the 12 power on the left-hand side
indicates the positive square root of x2. For example, if x 3, we get
[(3)2]12 912 3,
which is not equal to 3. To write a simpler equivalent expression for (x 2)12, we
use absolute value as follows.
Square Root of x 2
(x 2)12 x for any real number x.
x 2 x . Both of these equations are
Note that (x 2)12 x is also written as identities.
It is also necessary to use absolute value when writing identities for other even
roots of expressions involving variables.
E X A M P L E
7
Using absolute value symbols with roots
Simplify each expression. Assume the variables represent any real numbers and use
absolute value symbols as necessary.
x9 13
b) a) (x8y4)14
8
Solution
a) Apply the power of a product rule to get the equation (x8y4)14 x 2y. The lefthand side is nonnegative for any choices of x and y, but the right-hand side is
negative when y is negative. So for any real values of x and y we have
(x8y4)14 x2 y .
9.2
Rational Exponents
(9–17)
483
b) Using the power of a quotient rule, we get
x9
8
13
x3
.
2
This equation is valid for every real number x, so no absolute value signs are
■
used.
Because there are no real even roots of negative numbers, the expressions
a12,
x34,
and
y16
are not real numbers if the variables have negative values. To simplify matters, we
sometimes assume the variables represent only positive numbers when we are
working with expressions involving variables with rational exponents. That way we
do not have to be concerned with undefined expressions and absolute value.
E X A M P L E
8
Expressions involving variables with rational exponents
Use the rules of exponents to simplify the following. Write your answers with
positive exponents. Assume all variables represent positive real numbers.
a12
b) 1
a) x23x43
a 4
x2 12
c) (x12y3)12
d) 1
y 3
Solution
a) x23x43 x63
Use the product rule to add the exponents.
x2
Reduce the exponent.
12
a
b) 1
a1214
Use the quotient rule to subtract the exponents.
a 4
a14
Simplify.
12 3 12
12 12 3 12
c) (x y ) (x ) (y )
Power of a product rule
14 32
x y
Power of a power rule
14
x
3
Definition of negative exponent
y 2
d) Because this expression is a negative power of a quotient, we can first find the
reciprocal of the quotient, then apply the power of a power rule:
x2
1
y 3
12
y13
x2
12
y16
x
1 1 1
3 2 6
WARM-UPS
True or false? Explain your answer.
3
1. 913 9
5
2. 853 83
3. (16)12 1612
6
5. 612 6
7. 212 212 412
4. 932 27
2
6. 12
212
2
8. 1614 2
9. 616 616 613
1
10. (28)34 26
■
484
(9–18)
9.2
Chapter 9
Radicals and Rational Exponents
EXERCISES
Reading and Writing After reading this section, write out
the answers to these questions. Use complete sentences.
1. How do we indicate an nth root using exponents?
37. 1632
38. 2532
39. (27)13
40. (8)43
2. How do we indicate the mth power of the nth root using
exponents?
41. (16)14
42. (100)32
3. What is the meaning of a negative rational exponent?
Use the rules of exponents to simplify each expression. See
Examples 5 and 6.
43. 313314
44. 212 213
13 13
45. 3 3
46. 514514
13
8
2723
47. 2
48.
8 3
27 13
4. Which rules of exponents hold for rational exponents?
5. In what order must you perform the operations indicated
by a negative rational exponent?
6. When is amn a real number?
Write each radical expression using exponent notation and
each exponential expression using radical notation. See
Example 1.
4
7. 7
3
8. cbs
9. 915
11. 5x
13. a12
10. 312
12. 3y
14. (b)15
Evaluate each expression. See Example 2.
15.
17.
19.
21.
22.
2512
(125)13
1614
(4)12
(16)14
16. 1612
18. (32)15
20. 813
Write each radical expression using exponent notation and
each exponential expression using radical notation. See
Example 3.
3
w7
23. 1
25. 3
10
2
34
24. a5
26.
a1
3
53
28. 6
29. (ab)32
30. (3m)15
Evaluate each expression. See Example 4.
33. 25
32
35. 2743
50. 914 934
51. 1812 212
53. (26)13
52. 812212
54. (310)15
55. (38)12
56. (36)13
57. (24)12
58. (54)12
34
59. 6
2
54
60. 6
3
12
12
Simplify each expression. Assume the variables represent any
real numbers and use absolute value as necessary. See Example 7.
61. (x 4)14
62. (y6)16
8 12
63. (a )
64. (b10)12
65. (y3)13
66. (w9)13
67. (9x6y2)12
81x12 14
69. 2
y0
68. (16a 8b 4)14
144a8 12
70. 9y18
Simplify. Assume all variables represent positive numbers.
Write answers with positive exponents only. See Example 8.
71. x12x14
72. y13y13
73. (x12 y)(x34y12)
74. (a12b13)(ab)
w 13
75. w3
77. (144x 16 )12
a12 4
79. b14
a12
76. a2
78. (125a 8)13
2a12 6
80. 1
b 3
2
27. w
31. 12523
49. 434 414
32. 100023
32
34. 16
36. 1634
Simplify each expression. Write your answers with positive
exponents. Assume that all variables represent positive real
numbers.
81. (92)12
82. (416 )12
9.2
83. 1634
84. 2532
85. 12543
86. 2723
87. 212214
88. 91912
89. 30.26 30.74
91. 3142714
8 23
93. 27
1 34
95. 16
90. 21.520.5
92. 323923
8 13
94. 27
9 12
96. 16
(9–19)
Rational Exponents
485
In Exercises 123–130, solve each problem.
123. Diagonal of a box. The length of the diagonal of a
box can be found from the formula
D (L2
W2
H 2)12,
where L, W, and H represent the length, width, and
height of the box, respectively. If the box is 12 inches
long, 4 inches wide, and 3 inches high, then what is the
length of the diagonal?
D
9 12
9 13
97. (9x )
98. (27x )
99. (3a23)3
100. (5x12)2
4 in.
3 in.
12 in.
FIGURE FOR EXERCISE 123
101. (a12b)12(ab12)
102. (m14n12)2(m2n3)12
103. (km12)3(k 3m5)12
104. (tv13)2(t 2 v3)12
Use a scientific calculator with a power key (xy ) to
find the decimal value of each expression. Round answers to four decimal places.
105. 213
106. 512
107. 2
108. (3)
109. 1024110
110. 77760.2
111. 80.33
112. 2890.5
12
113.
64
15,625
0.75V
r 13
.
Find the radius of a spherical tank that has a volume
of 32
cubic meters.
3
13
16
32
114. 243
r
35
Simplify each expression. Assume a and b are positive real
numbers and m and n are rational numbers.
116. bn2 bn3
115. am2 am4
am5
117. am3
bn4
118. bn3
119. (a1mb1n)mn
120. (am2bn3)6
a3mb6n 13
124. Radius of a sphere. The radius of a sphere is a function of its volume, given by the formula
121. 9
am
a3mb6n 13
122. a6mb9n
FIGURE FOR EXERCISE 124
125. Maximum sail area. According to the new International America’s Cup Class Rules, the maximum sail
area in square meters for a yacht in the America’s Cup
race is given by
S (13.0368
7.84D13 0.8L)2,
where D is the displacement in cubic meters (m3), and
L is the length in meters (m). (Scientific American,
May 1992). Find the maximum sail area for a boat
that has a displacement of 18.42 m3 and a length of
21.45 m.
486
(9–20)
Chapter 9
Radicals and Rational Exponents
S (m2)
128. Best bond fund. The top bond fund for 1997 in the
5-year category was GT Global High Income B. An investment of $10,000 in 1992 grew to $21,830.95 in
1997. Use the formula from the previous exercise to
find the 5-year average annual return for this fund.
D (m3)
L (m)
FIGURE FOR EXERCISE 125
126. Orbits of the planets. According to Kepler’s third law
of planetary motion, the average radius R of the orbit
of a planet around the sun is determined by R T 23,
where T is the number of years for one orbit and R is
measured in astronomical units or AUs (Windows to
the Universe, www.windows.umich.edu).
a) It takes Mars 1.881 years to make one orbit of the
sun. What is the average radius (in AUs) of the
orbit of Mars?
b) The average radius of the orbit of Saturn is 9.05 AU.
Use the accompanying graph to estimate the number of years it takes Saturn to make one orbit of the
sun.
129. Overdue loan payment. In 1777 a wealthy Pennsylvania merchant, Jacob DeHaven, lent $450,000 to the
Continental Congress to rescue the troops at Valley
Forge. The loan was not repaid. In 1990 DeHaven’s
descendants filed suit for $141.6 billion (New York
Times, May 27, 1990). What average annual rate of return were they using to calculate the value of the debt
after 213 years? (See Exercise 127.)
130. California growin’. The population of California
grew from 19.9 million in 1970 to 32.5 million in 2000
(U.S. Census Bureau, www.census.gov). Find the average annual rate of growth for that time period. (Use
the formula from Exercise 127 with P being the initial
population and S being the population n years later.)
California population
(millions of people)
35
Radius of orbit (AU)
10
8
R T 2/3
6
4
30
25
20
15
10
2
0
10
20
Time for one orbit
(years)
30
FIGURE FOR EXERCISE 130
FIGURE FOR EXERCISE 126
127. Best stock fund. The average annual return for an investment is given by the formula
S
r P
1970 1980 1990 2000
Year
1n
1,
where P is the initial investment and S is the amount it
is worth after n years. The top mutual fund for 1997 in
the 3-year category was Fidelity Select-Energy Services (Money Guide to Mutual Funds, 1998), in which
an investment of $10,000 grew to $31,895.06 from
1994 to 1997. Find the 3-year average annual return
for this fund.
GET TING MORE INVOLVED
131. Discussion. If we use the product rule to simplify
(1)12 (1)12, we get
(1)12 (1)12 (1)1 1.
If we use the power of a product rule, we get
(1)12 (1)12 (1 1)12 112 1.
Which of these computations is incorrect? Explain
your answer.
132. Discussion. Determine whether each equation is an
identity. Explain.
a) (w 2 x 2)12 w x b) (w 2x 2)12 wx c) (w 2x 2)12 w x