7.3
8,700 lb, a coefficient of lift of 2.81, and a wing surface
area of 200 ft2. Find the proper landing speed for this
plane. What is the landing speed in miles per hour (mph)?
114.1 ft/sec, 77.8 mph
112. Landing speed and weight. Because the gross weight of
the Piper Cheyenne depends on how much fuel and cargo
are on board, the proper landing speed (from Exercise 111)
is not always the same. The formula V 1
9
.46
L gives
the landing speed as a function of the gross weight only.
a) Find the landing speed if the gross weight is 7,000 lb.
b) What gross weight corresponds to a landing speed of
115 ft/sec?
a) 102.3 ft/sec b) 8,840 lb
Operations with Radicals
7.3
407
3
a) x2 x b) x3 x 4
4
2
c) x x
d) x4 x n
n
x x an identity?
For which values of n is 114. Cooperative learning. Work in a group to determine
whether each inequality is correct.
0.9
b) 1.0
1 1.01
a) 0.9
3
3
c) 0.9
9 0.99
d) 1.0
01 1.001
n
For which values of x and n is x x?
115. Discussion. If your test scores are 80 and 100, then the
arithmetic mean of your scores is 90. The geometric mean
of the scores is a number h such that
80
h
.
h
100
GET TING MORE INVOLVED
113. Cooperative learning. Work in a group to determine
whether each equation is an identity. Explain your
answers.
(7-21)
Are you better off with the arithmetic mean or the
geometric mean?
OPERATIONS WITH RADICALS
In this section we will use the ideas of Section 7.2 in performing arithmetic operations with radical expressions.
In this
section
●
Adding and Subtracting
Radicals
●
Multiplying Radicals
●
Conjugates
Adding and Subtracting Radicals
To find the sum of 2 and 3, we can use a calculator to get 2 1.414 and
3 1.732. (The symbol means “is approximately equal to.”) We can then add
the decimal numbers and get
3
1.414 1.732 3.146.
2
We cannot write an exact decimal form for 2
3
; the number 3.146 is an ap. To represent the exact value of 2
3, we just use
proximation of 2 3
. This form cannot be simplified any further. However, a sum of
the form 2 3
like radicals can be simplified. Like radicals are radicals that have the same index
and the same radicand.
52
, we can use the fact that 3x 5x 8x is
To simplify the sum 32
52 82. So
true for any value of x. Substituting 2 for x gives us 32
like radicals can be combined just as like terms are combined.
E X A M P L E
1
Adding and subtracting like radicals
Simplify the following expressions. Assume the variables represent positive
numbers.
4
4
45
b) w
6w
a) 35
3
3
3
3
c) 3
5
43 65
d) 36x 2x 6x x
Solution
4
4
4
a) 35
45
75
b) w
6w
5w
c) 3
5
43 65
33 75
Only like radicals
are combined.
3
3
3
3
3
3
d) 36x 2x 6x x 46x 3x
■
408
(7-22)
Chapter 7
Rational Exponents and Radicals
We may have to simplify the radicals to determine which ones can be
combined.
E X A M P L E
2
Simplifying radicals before combining
Perform the indicated operations. Assume the variables represent positive numbers.
1
1
8
b) 20
a) 8
5
c) 2
x3 4
x2 518
x3
d) 16
x4
y3 54
x4
y3
3
3
Solution
a) 8
1
8 4 2
9 2
22
32
Simplify each radical.
52
Add like radicals.
Note that 8
18 2
6.
1
5
1
1 5 5
b) 2
0 25
Because and 20 25
5 5 5
5
5
5
5 105
Use the LCD of 5.
5
5
115
5
3
2
c) 2
x 4
x 518
x3 x2 2
x 2x 5 9
x2 2x
x2
x 2x 15x 2x Simplify each radical.
16x2
x 2x
Add like radicals only.
calculator
close-up
Check that
8 + 18
= 52.
d) 16
x4
y3 54
x4
y3 8
x3
y3 2x 27
x3
y3 2x
3
3
2 xy2x 3xy2x Simplify each radical.
3
xy2x
3
3
3
3
3
3
■
Multiplying Radicals
We have already multiplied radicals, in Section 7.2 when we rationalized denomin
n
n
nators. The product rule for radicals, a b ab, allows multiplication of
radicals with the same index, such as
5,
5 3 1
2 5 10,
3
3
3
and
x2 x x3.
5
5
5
CAUTION
The product rule does not allow multiplication of radicals that
3
and 5.
have different indices. We cannot use the product rule to multiply 2
E X A M P L E
3
Multiplying radicals with the same index
Multiply and simplify the following expressions. Assume the variables represent
positive numbers.
43
b) 3a2 6
a
a) 56
c) 4 4
3
3
d)
4
x3
2
4
x2
4
7.3
helpful
hint
Students often write
15
15
22
5 15.
Although this is correct, you
should get used to the idea
that
15
15
15.
Because of the definition of a
square root, a
a
a for
any positive number a.
(7-23)
Operations with Radicals
409
Solution
a) 56
43
5 4 6 3
201
8
Product rule for radicals
20 32 18 9 2 32
602
2
b) 3a 6a
18a3 Product rule for radicals
9a2 2
a
3a2a Simplify.
3
3
3
c) 4 4 16
3
3
8 2 Simplify.
3
22
d)
4
x3
2
4
x2
4
x5
8
4
4
x 4 · x
4
8
4
x
x
4
8
4
4
x
x · 2
4
4
8 · 2
4
x2x
2
4
Product rule for radicals
Simplify
Rationalize the denominator.
8 2 16 2
4
4
4
■
We find a product such as 32
(42 3
) by using the distributive property
as we do when multiplying a monomial and a binomial. A product such as
(23 5)(33 25) can be found by using FOIL as we do for the product
of two binomials.
E X A M P L E
4
Multiplying radicals
Multiply and simplify.
(42
3
)
a) 32
c) (23
5 )(33 25 )
b) a (a a2 )
2
d) (3 x
9 )
3
3
3
Solution
a) 32
(42 3
) 32
42 32
3 Distributive property
Because 2 2 2
12 2 36
and 2 3 6
24 36
3
3
3
3
3
b) a (a a2 ) a 2 a 3 Distributive property
3
a 2 a
5 )(33 25 )
c) (23
F
O
I
L
23 33
23 25
5
33 5
25
18 415 315 10
5 Combine like radicals.
8 1
410
(7-24)
Chapter 7
Rational Exponents and Radicals
d) To square a sum, we use (a b)2 a 2 2ab b 2:
2
2
(3 x
9 ) 32 2 3x
9 (x
9 )
9 6x
9 x 9
9
x 6x
■
In the next example we multiply radicals that have different indices.
E X A M P L E
5
Solution
3
4
a) 2 2 213 214
2712
12
27
12
128
3
b) 2 3 213 312
226 336
6
6
22 33
6
22
33
6
108
calculator
close-up
Check that
2 2 12
8.
3
4
Multiplying radicals with different indices
Write each product as a single radical expression.
3
4
3
b) 2 3
a) 2 2
12
Write in exponential notation.
Product rule for exponents: 1 1 7
3
4
12
Write in radical notation.
Write in exponential notation.
Write the exponents with the LCD of 6.
Write in radical notation.
Product rule for radicals
22 33 4 27 108
■
Because the bases in 213 214 are identical, we can add the
exponents [Example 5(a)]. Because the bases in 226 336 are not the same, we cannot add the exponents [Example 5(b)]. Instead, we write each factor as a sixth root
and use the product rule for radicals.
CAUTION
Conjugates
helpful
hint
The word “conjugate” is used
in many contexts in mathematics. According to the
dictionary, conjugate means
joined together, especially as
in a pair.
E X A M P L E
6
Recall the special product rule (a b)(a b) a 2 b 2. The product of the sum
4 3 and the difference 4 3 can be found by using this rule:
(4 3 )(4 3 ) 42 (3 )2 16 3 13
and the irrational number 4 3
is
The product of the irrational number 4 3
are
the rational number 13. For this reason the expressions 4 3 and 4 3
called conjugates of one another. We will use conjugates in Section 7.4 to rationalize some denominators.
Multiplying conjugates
Find the products. Assume the variables represent positive real numbers.
a) (2 35 )(2 35 )
b) (3 2 )(3 2 )
c) (2x y )(2x y )
7.3
Operations with Radicals
Solution
2
a) (2 35 )(2 35 ) 22 (35 ) (a b)(a b) a 2 b 2
4 45
(35 )2 9 5 45
41
b) (3 2 )(3 2 ) 3 2
1
c) (2x y )(2x y ) 2x y
WARM-UPS
411
■
True or false? Explain.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
7.3
(7-25)
3 3
6
False
2
32
True
8
33
63
False
23
3
3
2 2 2 False
32
61
0 True
25
35
51
0 False
25
(3 2
) 6 2 True
2
2 26
False
1
(2 3 )2 2 3 False
(3 2 )(3 2 ) 1 True
EXERCISES
Reading and Writing After reading this section, write out the
answers to these questions. Use complete sentences.
1. What are like radicals?
Like radicals are radicals with the same index and the same
radicand.
2. How do we combine like radicals?
Like radicals are combined using the distributive property
just as we combine like terms.
3. Does the product rule allow multiplication of unlike
radicals?
In the product rule the radicals must have the same index
but do not have to have the same radicand.
4. How do we multiply radicals of different indices?
To multiply radicals of different indices, we convert them
to equivalent radicals with the same index.
All variables in the following exercises represent positive
numbers.
Simplify the sums and differences. Give exact answers. See
Example 1.
5. 3
23 3
6. 5
35 25
7. 57
x 47x 97x
8. 36
a 76a 106a
3
3
3
9. 22 32 52
3
3
3
10. 4 44 54
11. 3
5 33 5 43 25
12. 2
53 72
93 62 43
3
3
3
3
3
13. 2 x 2 4x 5x
3
3
3
3
3
3
14. 5y 45y x x 35y 2x
3
3
3
15. x 2
x x 2x 2x
3
3
3
16. ab a 5a ab 2ab 6a
Simplify each expression. Give exact answers. See Example 2.
17. 8
28 22 27
23 26
18. 1
2 24
2
19. 2
8
20. 2
0 125
35
2
21. 2
2
32
2
3
22. 3
3
23
3
23. 8
0 5
1
215
5
412
(7-26)
24. 3
2 Chapter 7
Rational Exponents and Radicals
Find the product of each pair of conjugates. See Example 6.
67. (3 2)(3 2) 1
92
2
1
2
x 18
x 2 50
x 2 20
x 3 x5x 2x2
25. 45
3
26. 12
x 18x 300
x 98x 42x 8x 3x
5
5
53
27. 24 81
3
3
3
3
3
29. 48 243
3
30. 64 2
5
4
4
5
73.
32
74.
31. 54
t 4
y 3 16
t 4
y 3 ty2t
3
3
3
75.
32. 2000w
z 16w
z 8z2w
z 3
3
2 5
3
2 5
2 2
Simplify the products. Give exact answers. See Examples 3 and 4.
33. 3
5 15
34. 5
7 35
35. 25
310
70.
72.
4
5
68.
69.
71.
73
28. 24 375
3
2
76.
(7 3 )(7 3 ) 46
(5 2 )(5 2 ) 3
(6 5 )(6 5) 1
(25 1)(25 1) 19
(32 4)(32 4) 2
(32 5 )(32 5 ) 13
(23 7 )(23 7 ) 5
(5 3x )(5 3x ) 25 9x
(4y 3z )(4y 3z) 16y 9z
Simplify each expression.
77. 300 3 113
78. 50 2
62
302
36. (32 )(410 ) 245
79. 25
56
37. 27
a 32a
80. 36 510
6a14
38. 25
c 55
39. 9 27
4
50c
4
40. 5 100
3
41. (23 )
82.
3
42. (42 )
3
4x 2
3
3
32
2
2x3x
3
4
3x 22x
86. 2
t 10
t
2t 4 5t
4
1
1
1
87. 2 8 18
4
3
15m
x 6
x
85. 3
5
3
2x2
3
4
45. 23
(6 33 ) 62 18
1
88. 3
1
3 3
46. 25 (3 35 ) 215 30
89.
0 2) 52 25
47. 5 (1
90.
48. 6 (15 1) 310 6
2 2 82
91. 3
5
15
49. 3t(9t t 2 ) 3
t 2 t3
3
3
3
8 7
3
2
3
4x
x
5
44. 5
4
2x
5 2x
5
43.
(3 27 )(7 2)
(2 7 )(7 2)
5m
84. 3m
12
2
3015
4w
16w
83. 4w
54
3
2
81.
33
4
1030
3
3
52
12
3
3
(25 2 )(35 2 )
(32 3 )(22 33 )
51. (3 2)(3 5) 7 33
2 3 52 43
92. 4
5
20
52. (5 2)(5 6) 7 45
93.
50. 2(12x 2x ) 23x 4x
3
3
3
3
3
53. (11 3)(11 3) 2
94.
54. (2 5)(2 5) 27 102
95.
55. (25 7)(25 4) 8 65
96.
56. (26 3)(26 4) 12 26
97.
57. (23 6 )(3 26 ) 6 92
98.
58. (33 2 )(2 3 ) 26 7
99.
Write each product as a single radical expression. See Example 5.
3
6
4
4
59. 3 3 35
60. 3 3 27
100.
101. 4
w
9w
w
102. 10m
16m
6m
5
62. 2 2
2
63. 2 5
500
64. 6 2
864
103. 2a3 3a3 2a4a
65. 2 3
432
66. 3 2
648
104. 5w
y 7w
y 6w
y
4
3
3
4
12
7
6
12
3
5
3
3
4
15
8
6
12
3 76
(5 22 )(5 22 ) 17
(3 27 )(3 27 ) 19
(3 x )2 9 6x x
(1 x )2 1 2x x
(5x 3)2 25x 30x 9
(3a 2)2 9a 12a 4
2
(1 x
2 ) x 3 2x
2
2
(x
1 1) x 2x
1
61. 5 5
3
28 10
2
2
105. 5
0a 18a 2a
aa
2
4wy
72a
7.4
More Operations with Radicals
3x 2x
108. 8
x 3 50
x 3 x2x
109.
110.
111. 16
x 5x54x
13x2x
112. 3
x
y 24
x
y
xy 23y
x 2
3
113.
3
4
3
5 7
5x x
5
3
3
5 7
y7 y22x2y
4x
2x
5
115.
––
√ 6 ft
6x2x
(a a 3)(a a 3) a a6
(w
z 2y 4)(w
z 2y 4) wz 4y8
3
3
—–
√12 ft
3
FIGURE FOR EXERCISE 121
3
3
114.
3
x2 5
5
4
122. Area of a triangle. Find the exact area of a triangle with
a base of 30
meters and a height of 6 meters.
35 square meters (m2)
29z
3z
4
16
3
9z
116. a3 (a a5 ) a a2
4
4
4
117. 2x 2x 32
x 5 118. 2m
2n
3
413
–
√ 3 ft
106. 200z 128z 8z 162z
107. x 5 2x
x3
(7-27)
6
3
4
––
√6 m
128m
4n3
12
In Exercises 119–122, solve each problem.
119. Area of a rectangle. Find the exact area of a rectangle
feet.
that has a length of 6 feet and a width of 3
32
square feet (ft2)
120. Volume of a cube. Find the exact volume of a cube with
sides of length 3 meters.
33
cubic meters (m3)
––
√3 m
––
√3 m
FIGURE FOR EXERCISE 122
GET TING MORE INVOLVED
123. Discussion. Is a
b a
b for all values of
a and b?
No
124. Discussion. Which of the following equations are identities? Explain your answers.
a) 9x 3x
b) 9
x 3 x
c) x
4 x 2
x x
d) 4
2
a and d
125. Exploration. Because 3 is the square of 3, a binomial
such as y 2 3 is a difference of two squares.
a) Factor y 2 3 and 2a 2 7 using radicals.
b) Use factoring with radicals to solve the equations
x 2 8 0 and 3y 2 11 0.
c) Assuming a and b are positive real numbers, solve the
equations x 2 a 0 and ax 2 b 0.
––
√3 m
FIGURE FOR EXERCISE 120
121. Area of a trapezoid. Find the exact area of a trapezoid
with a height of 6 feet and bases of 3 feet and
1
2 feet.
92 2
ft
2
7.4
—–
√30 m
MORE OPERATIONS WITH RADICALS
In this section you will continue studying operations with radicals. We learn to rationalize some denominators that are different from those rationalized in Section 7.2.
In this
section
●
Dividing Radicals
●
Rationalizing the
Denominator
●
Powers of Radical
Expressions
Dividing Radicals
In Section 7.3 you learned how to add, subtract, and multiply radical expressions.
To divide two radical expressions, simply write the quotient as a ratio and then
simplify, as we did in Section 7.2. In general, we have
a
n
n
a b n
b
n
b,
n
a