Download Chapter 8 Rational Exponents, Radicals, and Complex Numbers Exercise Set 8.1

Chapter 8
56.
Rational Exponents, Radicals, and
Complex Numbers
With the radical symbol.
4.
The nonnegative root of a number.
2
66. 3.330
68. –2.962
2
x means the nonnegative square root of x ,
even if x is negative.
8.
±8
12. ±20
14. ±17
60. –6.403
64. –4.362
6.
10. ±9
58. 3.464
62. 2.759
Exercise Set 8.1
2.
1
2
70. 2.283
72. 2.147
74. r 4
76. 9t
78. 11x 4 y 5
16. 7
80. 0.9r 2 s 7
18. not a real number
82. n 2
20. 10
84. 4u 4 t 3
22. s10
24. 1.4
26. not a real number
28. 0.13
30.
8
13
32. 5
2
86. 3r 5 s
88. 0.3r 4
90. x3
92. 3t 5
94. 3x3
96. st 3
34. 5
98. 3 t 3
36. 6
100. k 3
38. 5
102. 2 x5
40. 3
42. not a real number
104. 5x 2
44. 5
106.
x 2 2
48. 2
108.
n 53
50. 5
110.
11
52. 3
112.
7
46. 3
54. 4
5
114.
^ x x u 4` , or > 4, g
50
Chapter 8 Rational Exponents, Radicals, and Complex Numbers
^ x x e 3` , or g,3@
116.
118. a)
12.
3
216
14.
3
125
16.
8
w
6
5
121z 6
18.
11z 3
20. 22 a
b)
^ x x u 3` , or > 3, g
22.
n8
36
24.
16 120. a)
b)
^ x x u 3` , or > 3, g
30.
3
4
26. 28.
n4
6
16 124. } 4.916 sec.
32.
126. } 31.305 mph
5
32
5
243
4
3
27
1
1
16
8
4
3
122. } 13.282 m/sec.
8
7
m5
34. 6 6 a5
128. } 107.7 ft.
130. 17 N
36.
132. a) ≈416,438
b) ≈555,209
1
3
216
¨ 1 ·
38. © 5
¸
ª 32 ¹
Puzzle Problem
9,9,16 (The ages cannot be 4,4,81 because a 41year-old mother cannot have an 81-year-old
child or 4, 9, 36 because a 41-year-old mother
cannot have a 36-year-old child.)
40.
7
3
5r 25
42. 421/5
44. r 5/7
Exercise Set 8.2
46. 62/7
2.
5
4.
No restrictions would be placed on a because
with an odd index, the radicand can be positive
or negative.
6.
Yes, because
31/ 3 ™ 21/ 5
8.
64
10. 64
35/15 ™ 23/15
8
8
48. 8n 9/7
50. 86/5
52. m8/3
3 ™ 2 5
3 1/15
19441/15 .
54.
5w 35/7
56.
3r 69/5
58. n6/7
1
36
2
1
8
Instructor’s Solutions Manual
60. n1/6
62. r 25/6
64. 15 p
51
114.
6
y7
116.
6
z
118.
6
2000
120.
12
1024
122.
15
m
124.
20
z
5/9
66. 32c
23/10
5/9
68. 3
70.
1
y1/3
72. x1/8
Exercise Set 8.3
74. m1/10
2.
The radicand, 28, has the perfect square 4 as a
factor. So 28 2 ™ 7 .
4.
The radicand, x5 , has the perfect cube x3 as a
factor.
80. 36u1/3
6.
12
82. 42 y 5/6
8.
4 y2
84. r 3
10. 10u 2 v3
86. n3/2
12.
66
88. m3/5
14.
17 y
90. 256a 2 b12
16. 4
92. 8x1/2 y1/4
18.
3
12m 2
94. 25v 2
20.
4
35
96. 6x1/4
22. r 4 147
76.
1
v 6/5
78. b3/2
98.
7
100.
3
24.
10
102.
12a 2 b3
26. m 5 24m 4
y
28.
104.
4
5
6
n3
2
106.
3
yz
108.
5
ab4
110.
12
y11
112.
15
7a 4 b5
30.
55
6
32.
7a
3b
34.
x 22
36.
7
9
15
9
52
Chapter 8 Rational Exponents, Radicals, and Complex Numbers
38. 9
92. x 4 y 3 y
7
40.
42.
44.
94. 60c3 6
3
7
v
3
3
5a
96. 90 2
98. 3 2
100. 10 7
2r 2
46. 3
102. mn m
4
48.
3x 2
3y
104. 8n 2
106. 6a 2 6a
50. 4 6
108. 24r 2 s 3 3r
52. 6 5
54. 20 3
56. 21 3
110.
4 3
5
112.
c6
6
114.
5 y3 2
9
116.
5
2
12
58. d 2 d
3
60. a b
62. p 2 q 4 r 4
64. a 7 b 4 ab
66. 30d 3d
Exercise Set 8.4
68. 4 3 2
2.
70. b
33
b
2
72. m 4 n3 3 m
4.
76. 20 3 2
78. 3 4 2
80. 6a5 4 3a 3
82. 2n
2
5n
7
a b { a b because the order of
operations is not the same. a b indicates
that the radicals must be evaluated first and then
the addition. a b indicates that the addition
must be evaluated first and then the radical.
Example: 36 64 { 36 64 since
6 8 { 10 .
74. 2h 4 3 6h 2
35
3x 2 x 5 x,3 2 2 2 5 2 . In both cases,
the coefficients of like terms are added.
6.
−9 5
8.
12 y
3
2 2 5
84. a bc a b c
10. 6 7 − 9 11
86. 5 3
12. 4n 2n + 4m 5m
88. 30 14
14. 3 y 3 3
90. m5 m
16. −6 y 3 4 8 y
Instructor’s Solutions Manual
53
18. Cannot combine because the radicals are not
like.
76. 4
20. −2 6
80. 25 − y
22. 8 3x
82. 2
24. −10 5
84. a − b
78. −2
86. 37
26. 19 7
28. −28 y 2 2 y
30. 3 5
88. 62
90. 14 3
92. −6 5
32. 2 5 − 12 3
94. 6 3
3
34. 5 3
96. −32 3
36. 33a3 3 4a
38.
14 y 2 4
3y
98. 7 5 + 21
2
100. a)
40. 4 5 − 5
( 232 + 8 2 ) in.
b) 243.3 in.
42. 6 + 3 2
44.
35 − 21 2
46. 36 y + 24 y 5
Exercise Set 8.5
The denominator in
4.
Multiply the fraction by a 1 whose numerator
and denominator are the conjugate of the
denominator.
48. 12 − 2 3 + 6 5 − 15
50. 10 + 3 a − a
52. 14 − 14 10 − 6 5 + 30 2
54.
10 + 35 + 2 + 14
56. 2a + 9 ab + 4b
58. 16 10 + 48 5 − 4 15 − 6 30
6.
7
7
8.
5 3
6
10.
8 3
3
12.
22
6
14.
x 6
8
16.
6
6
60. 6m + 10 mn − 4n
62. 3 3 3 + 3 3 9 − 10
64. 2 − 3 15 + 3 225
66. r − 27
68. 16 + 6 7
70. 27 − 10 2
72. 29 + 12 5
74. 182 − 80 3
3
is a rational number,
3
1
.
whereas it is irrational in
3
2.
54
Chapter 8 Rational Exponents, Radicals, and Complex Numbers
56. 7 7 − 7 6
18.
11 7b
7b
20.
30rs
6s
60.
5+ 5
4
22.
4 7a
a
62.
−10 − 8 5
11
24.
26.
58. −1 − 7
5
3
64. 4 2 − 2 6
24 2 − 2 6
47
4x − 4 x
68.
x −1
5 b
b
66.
3
28. 3a3 6a
30. Mistake: Multiplied
Correct:
21
3
3
32.
7 25
5
3 ™ 3 incorrectly.
70.
2 mn + 6m
n − 9m
72.
2h 7 − 14hk
2h − k
74.
3
34.
6
2
76.
36. 3 3 3
38.
p 3 q2
q
3
40.
80.
mn
n
3 3 9a 2
42.
a
3
44.
4
5y
y
4
50.
6 8x
x
82.
84.
20b
4b
74 9
7 3
or
46.
3
3
48.
78.
2
52. 3 5 − 6
54. 8 + 2 15
7
3 7
7y
3 7y
5t
4 5t
7
12 − 4 2
2 x − 49
3 2 x − 21
6
20 k + 5 10k
86. a)
10 + 2
4
b)
6+ 2
17
c)
4 3+ 2
23
88. a) The graphs are identical. The functions are
identical.
b) f x g x
Instructor’s Solutions Manual
90. a) t =
b) t =
55
46. 6
h
4
h
48. −3 ( −7 is an extraneous solution.)
50. −2, − 1
4 h
52. 0, 4
92. a) s =
b)
2A 3
3
54.
10 2 3
meters
3
56. 7
58. −5, 4
c) ≈ 6.204 m
60. 0, 4
2 Em
m
94. a)
62. 2
64. 2 (
b) 4, 000, 000 3 m/sec.
)
96. a) 12 − 4 3 cm
b) 5.1 cm
68. Mistake: the radical
correctly.
Correct: x = 13
Exercise Set 8.6
2.
4.
6.
8.
A solution that does not satisfy the original
equation.
( a)
2
1
is an extraneous solution.)
4
66. Mistake: You cannot take the principal square
root of a number and get a negative.
Correct: No real-number solution.
c) 6,928,203.2 m/sec.
(
1 5
( is an extraneous solution.)
3 3
= a ⋅ a = a2 = a
Answers may vary. One example is
x + 4 = 2 x + 1 . The radicals must be totally
eliminated.
25
x + 3 was not squared
70. 88.2 m
72. 0.272 m
74. 4 ft
76.
1
foot
4
78. 146.94 ft.
80. 229.59 ft.
82. 6 N
10. no real-number solution
84. 4 N
12. 64
14. −125
16. 11
18. −7
20. 2
22. −3
86. a)
x
y
0
2
1
3
4
4
9
5
16
6
25
7
b)
24. no real-number solution
26. 62
28. −34
30. 35
32. 21
34. 4
36. −121
42. No real-number solution ( −3 is an extraneous
solution.)
c) No. The x-values must be 0 or positive
because real square roots exist only when x u 0 .
The y-values must be 0 or positive because, by
definition, the principal square root is 0 or
positive, and adding 2 to those results still gives
positive results.
44. 4
d) Yes, because it passes the vertical line test.
38. no real-number solution
40. 5
56
Chapter 8 Rational Exponents, Radicals, and Complex Numbers
88. The graph extends below the x-axis; that is, it
reflects the graph across the x-axis.
90. The graph begins at (0, 3), extending downward,
and is steeper than the graph of y = − x .
Collaborative Exercises
48. 22 − 3i
50. 66 + 27i
52. 97
54. 16 − 30i
56. 6i
6i
7
2.
Increase the length to 0.993 m.
58. −
3.
The length increases.
60. −4i
62.
Exercise Set 8.7
1
−i
3
2.
Yes, every real number can be expressed as
a 0i .
1
64. − − i
3
4.
We add complex numbers just like we add
polynomials—by combining like terms.
66.
6.
No, because it is not in the form a bi .
8.
9i
30 5
− i
37 37
68. −
6 10
+ i
17 17
10. i 10
70. 1 + i
12. 2i 3
72.
11 13
+
i
29 29
74.
64 33
+ i
85 85
14. 4i 2
16. 6i 2
76. i
18. 4i 5
78. −i
20. 3i 6
80. i
22. 9i 10
82. 1
24. 4 + 2i
26.
4 i 7 6i 84. 1
11 5i
86. −1
28. 9 − i
88. −i
30. 19 + 8i
90. 1
32. 6 − 16i
34. 0
36. −1 + 11i
38. −22 − 9i
40. −45
42. 28
44. 6 + 54i
46. −56 − 35i