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8
Powers and
Roots
ailing—the very word conjures up images of warm summer
breezes, sparkling blue water, and white billowing sails. But to
boat builders, sailing is a serious business. Yacht designers know
that the ocean is a dangerous and unforgiving place. It is their job to build
boats that are not only fast, comfortable, and fun, but capable of withstanding the punishment inflicted by the wind and waves.
Designing sailboats is a technical balancing act. A good boat has just
the right combination of length, width (or beam), sail area, and displacement. A boat can always be made faster by increasing the sail
area, but too much sail area increases the chance of capsize. Making
the boat wider decreases the chance of capsize, but causes more resistance from the water and slows down the boat.
There are four ratios commonly used to measure performance
and safety for a yacht: ballast-displacement ratio, displacementlength ratio, sail area-displacement ratio, and capsize screening
value. The formulas for these ratios involve powers and
roots, which is the subject of this chapter. In Exercise 71 of
Section 8.5 you will find the sail area-displacement ratio
for a sailboat.
Sail area-displacement ratio (r)
S
20
Sail area 810 ft2
18
16
14
12
10
20
22
24 26 28 30
Displacement
(thousands of pounds)
414
(8-2)
Chapter 8
Powers and Roots
8.1
In this
section
●
Fundamentals
●
Roots and Variables
●
Product Rule for Radicals
●
Quotient Rule for Radicals
ROOTS, RADICALS, AND RULES
In Section 4.6 you learned the basic facts about powers. In this section you will
study roots and see how powers and roots are related.
Fundamentals
We use the idea of roots to reverse powers. Because 32 9 and (3)2 9, both 3
and 3 are square roots of 9. Because 24 16 and (2)4 16, both 2 and 2 are
fourth roots of 16. Because 23 8 and (2)3 8, there is only one real cube root
of 8 and only one real cube root of 8. The cube root of 8 is 2 and the cube root
of 8 is 2.
nth Roots
If a bn for a positive integer n, then b is an nth root of a. If a b2, then b
is a square root of a. If a b3, then b is the cube root of a.
If n is a positive even integer and a is positive, then there are two real nth roots
of a. We call these roots even roots. The positive even root of a positive number is
called the principal root. The principal square root of 9 is 3 and the principal fourth
root of 16 is 2 and these roots are even roots.
If n is a positive odd integer and a is any real number, there is only one real nth
root of a. We call that root an odd root. Because 25 32, the fifth root of 32 is 2
and 2 is an odd root.
We use the radical symbol to signify roots.
n
a
n
If n is a positive even integer and a is positive, then a denotes the principal
nth root of a.
n
If n is a positive odd integer, thenn a denotes the nth root of a.
If n is any positive integer, then 0 0.
n
n
We read a as “the nth root of a.” In the notation a, n is the index of the
radical and a is the radicand. For square roots the index is omitted, and we simply
.
write a
E X A M P L E
1
Evaluating radical expressions
Find the following roots:
3
5
b) 27
a) 2
c) 64
Solution
a) Because 52 25, 2
5 5.
3
b) Because (3)3 27, 27 3.
6
c) Because 26 64, 64 2.
2, 4 (4
) 2.
d) Because 4
6
d) 4
■
8.1
Roots, Radicals, and Rules
(8-3)
415
calculator
In radical notation, 4
represents the principal square root of
4, so 4 2. Note that 2 is also a square root of 4, but 4 2.
close-up
Note that even roots of negative numbers are omitted from the definition of nth
roots because even powers of real numbers are never negative. So no real number
can be an even root of a negative number. Expressions such as
We can use the radical symbol
to find a square root on a
graphing calculator, but for
other roots we use the xth
root symbol as shown. The
xth root symbol is in the
MATH menu.
CAUTION
9, 81, and
64
4
6
are not real numbers. Square roots of negative numbers will be discussed in Section 9.5 when we discuss the imaginary numbers.
Roots and Variables
Consider the result of squaring a power of x:
(x1)2 x 2, (x 2)2 x4, (x 3)2 x 6, and
(x4)2 x8.
When a power of x is squared, the exponent is multiplied by 2. So any even power
of x is a perfect square.
Perfect Squares
The following expressions are perfect squares:
x 2,
calculator
close-up
A calculator can provide numerical support for this discussion of roots. Note that
(
3)
2 3 not 3 because
2
x x when x is negative.
Note also that the calculator
will not evaluate 32 be2
cause 3 9 .
x4,
x 6,
x8,
x 10,
x12,
...
Since taking a square root reverses the operation of squaring, the square root of an
even power of x is found by dividing the exponent by 2. Provided x is nonnegative
(see Caution below), we have:
x2 x1 x,
x4 x 2, x6 x 3, and
x8 x 4.
If x is negative, equations like x2 x and x6 x 3 are not
correct because the radical represents the nonnegative square root but x and x 3 are
negative. That is why we assume x is nonnegative.
CAUTION
If a power of x is cubed, the exponent is multiplied by 3:
(x1)3 x 3, (x 2)3 x6, (x 3)3 x 9, and
(x 4)3 x12.
So if the exponent is a multiple of 3, we have a perfect cube.
Perfect Cubes
The following expressions are perfect cubes:
x3,
x6, x 9,
x12,
x15,
...
Since the cube root reverses the operation of cubing, the cube root of any of these
perfect cubes is found by dividing the exponent by 3:
x 3 x 1 x,
3
x 6 x 2, x 9 x 3, and
3
3
x 12 x 4.
3
If the exponent is divisible by 4, we have a perfect fourth power, and so on.
416
(8-4)
Chapter 8
E X A M P L E
2
Powers and Roots
Roots of exponential expressions
Find each root. Assume that all variables represent nonnegative real numbers.
3 18
5
b) t
c) s30
a) x22
Solution
a) x22 x11 because (x11)2 x 22.
3 18
t t 6 because (t 6)3 t18.
b) 5
s 30 s 6 because one-fifth of 30 is 6.
c) ■
Product Rule for Radicals
Consider the expression 2
3
. If we square this product, we get
calculator
(2 3 )2 (2)2(3)2
Power of a product rule
23
close-up
(2 )2 2 and (3 )2 3
6.
You can illustrate the product
rule for radicals with a
calculator.
The number 6
is the unique positive number whose square is 6. Because we
3
and obtained 6, we must have 6
2
3
. This examsquared 2
ple illustrates the product rule for radicals.
Product Rule for Radicals
The nth root of a product is equal to the product of the nth roots. In symbols,
ab a b,
n
n
n
provided all of these roots are real numbers.
E X A M P L E
3
calculator
close-up
You can illustrate the quotient rule for radicals with a
calculator.
Using the product rule for radicals
Simplify each radical. Assume that all variables represent positive real numbers.
y
b) 3
y8
a) 4
Solution
a) 4
y 4
y
2y
b) 3
y8 3 y8
3
y4
y4 3
Product rule for radicals
Simplify.
Product rule for radicals
y8 y4
A radical is usually written last in a product.
Quotient Rule for Radicals
Because 2
3
6
, we have 6
3
2
, or
2 63 .63
This example illustrates the quotient rule for radicals.
■
8.1
Roots, Radicals, and Rules
(8-5)
417
Quotient Rule for Radicals
The nth root of a quotient is equal to the quotient of the nth roots. In symbols,
a a
n ,
b b
provided that all of these roots are real numbers and b 0.
n
n
In the next example we use the quotient rule to simplify radical expressions.
E X A M P L E
4
Using the quotient rule for radicals
Simplify each radical. Assume that all variables represent positive real numbers.
21
t
3 x
a) b) 6
9
y
Solution
t
t
a) 9 9
t
3
b)
3
Quotient rule for radicals
3
x21 x21
6 3 y
y6
7
x
2
y
M A T H
Quotient rule for radicals
■
A T
W O R K
3–2–1–Blast off! Joseph Bursavich watches each
Space Shuttle mission with particular attention. He
is a Senior Computer Systems Designer for Martin
Marietta, writing programs that support the building
COMPUTER
of the external tank that is the structural backbone of
SYSTEMS
the Space Shuttle for NASA. Each tank, made up of
DESIGNER
three major parts, is 157 feet long and takes almost
two years to build. To date, 74 tanks have been completed, and each has done its job
in carrying cargo into space.
Currently, Mr. Bursavich supports a team whose objective is to develop a
superlightweight tank made of a mixture of aluminum and other metals. Reducing
the weight of the tank is vital to the space station program because a lighter tank
means that each mission can carry a greater payload into space. Because of economic considerations, as many as ten external tanks might be produced at one time.
Mr. Bursavich writes programs to determine the economic order quantity (EOQ) for
components of the external tank. The EOQ depends on setup costs, labor costs, the
quantity of the component to be used in one year, the cost of holding stock for one
year, and maintenance costs.
In Exercise 83 of this section you will use the formula that Mr. Bursavich uses
to determine the EOQ for component parts of the tanks.
418
(8-6)
Chapter 8
WARM-UPS
Powers and Roots
True or false? Explain your answer.
1. 2 2
2
3
3. 27 3
False
True
4. 25 5
False
True
7. 29 23
False
6. 9
3
9. If w 0, then w
w.
2. What is a principal root?
The principal root is the positive even root of a positive
number.
3. What is the difference between an even root and an odd
root?
If bn a, then b is an even root provided n is even or an
odd root provided n is odd.
4. What symbol is used to indicate an nth root?
n
The nth root of a is written as a.
5. What is the product rule for radicals?
n
n
n
The product rule for radicals says that a b ab
provided all of these roots are real.
6. What is the quotient rule for radicals?
n
n
n
The quotient rule for radicals says that ab ab
provided all of these roots are real.
For all of the exercises in this section assume that all variables
represent positive real numbers.
Find each root. See Example 1.
7. 36 6
8. 4
9 7
9. 32
5
11. 1000
10. 81
10
4
13. 16
True
12
True
3
1
2 m
31. m
32. m
6 m3
33. y15 y3
34. m
8 m2
35. y15 y5
36. m
8 m4
37. m
3 m
38. x4
x
39. 3 27
40. 4
4
41. 2
42. 2
5
4
4
21. 81
3
3
6
6
10
24. 128
25. 32
27. 100
2
10
3
26. 125
28. 36
3
32
43. 5 125
3
45. 10
2
5
6
99
10
233
44. 10
1,000,000
46. 10
109
3
9
20
18
18
10
Use the product rule for radicals to simplify each expression.
See Example 3.
y 3y
47. 9
48. 16n
4n
49. 4a 2a
50. 36n
6n
x
y
51. 2
52. w
t w3t
m65
54. 7
z16 z87
2
4 2
2
6 2
xy
12
53. 5m
55. 8y 2y
56. 27
z2
57. 27w
3 3w
58. 125m
6
3
3
3
z2
3
3
3
5m2
3
60. 81w
3w
4
4
4
5a3y2
62. 27
z3w
15
3zw5
3
Simplify each radical. See Example 4.
t
63.
4
65.
16
67.
69.
y8x
1
7
2
2
61. 125a9y6
Not a real number
5
4
2
Not a real number
22. 64
4
3
0
20. 1
1
23. 64
Find each root. See Example 2.
59. 16s 2s
12. 16
3
Not a real number
3
4
18. 0
Not a real number
144
30. 16. 1
1
0
6
4
29. 50
4
Not a real number 14. 1 Not a real number
17. 1
3
4
2
3
19. 1
False
10. If t 0, then t t 3.
4
EXERCISES
Reading and Writing After reading this section, write out the
answers to these questions. Use complete sentences.
1. How do you know if b is an nth root of a?
If bn a, then b is an nth root of a.
15. 0
True
8. 17 1
7 289
2
8. 1
3
2. 2 2 2
5. 16 2
4
3
True
t
2
625
3
25
4
t t
8 2
64.
36
w
w
66.
144
9
1
4
68.
a
3
70.
12070y0 3
6
3
3
2x2
y
3
3
a
27
6
3
36
3y12
10
8.1
71.
49a
73.
6
4
y
16
499ab
5w
74.
81
2a3
3
2
72.
y
2
4
4
4
5w
3
4
3
76. 7
0.914
5 2
77. 3 4
1.610
2 3
78. 1 5
0.257
79. 7.1
2
(1
4.2
)(
3)
200
100
0
6.001
0
10
20
30
Altitude
(thousands of feet)
(
42)(
0.2
) 3.256
80. 32
8 (
8
)2
(1
4.3
)(
6.2
)
82. 2(1.3)
40
FIGURE FOR EXERCISE 85
0.769
6.850
Solve each problem.
83. Economic order quantity. When a part is needed for a
space shuttle external fuel tank, Joseph Bursavich at Martin
Marietta determines the most economic order quantity E by
using the formula E 2AS, where A is the quantity that
I the plant will use in one year, S is the cost of setup for
making the part, and I is the cost of holding one unit in
stock for one year. Find the most economic order quantity
if S $5290, A 20, and I $100. 46
84. Diagonal of a box. The length of the diagonal D of the
box shown in the figure can be found from the formula
D L
2
W2
H2,
where L, W, and H represent the length, width, and height
of the box, respectively. If the box has length 6 inches,
width 4 inches, and height 3 inches, then what is the length
of the diagonal to the nearest tenth of an inch? 7.8 in.
86. Sailing speed. To find the maximum speed in knots (nautical miles per hour) for a sailboat, sailors use the formula M 1.3w
, where w is the length of the waterline
in feet.
a) If the length of the waterline for the sloop John B. is
38 feet, then what is the maximum speed for the John B.?
b) Use the accompanying graph to estimate the length of
the waterline for a boat for which the maximum speed is
6 knots.
a) 8.0 knots
b) 20 ft
10
Maximum speed (knots)
3 32
(1
4)(
2.9
)
81. 2
419
300
View (miles)
3.968
(8-7)
a) Use the formula to find the view to the nearest mile from
an altitude of 35,000 feet.
b) Use the accompanying graph to estimate the altitude of
an airplane from which the view is 100 miles.
a) 228 mi
b) 7000 ft
3a
2
7b
Use a calculator to find the approximate value of each
expression to three decimal places.
5
75. 3
Roots, Radicals, and Rules
8
6
4
2
0
0
10
20 30 40
Waterline (feet)
50
FIGURE FOR EXERCISE 86
D
GET TING MORE INVOLVED
6 in.
3 in.
87. Discussion. Determine whether each equation is correct.
5
)2 5
a) (
4 in.
FIGURE FOR EXERCISE 84
85. Buena vista. The formula V 1.22A
gives the view in
miles from horizon to horizon at an altitude of A feet (Delta
Airlines brochure).
b) (
2
)3 2
3
3
)4 3
d) (7
)5 7
c) (
a) no
b) yes
c) no
d) yes
n
88. Writing. If x is a negative number and x n x, then what
can you say about n? Explain your answer.
n is odd
4
5