Section 10.6 Radical Equations
45. 232
l
t
π
232
tl
π
46.
FR
vm
47. Let x = the number.
54 2
xx
22
54 2
xx
Check
1
x
:
51 4 1 2
54 1
11
074
xx
70
7
x
x
or 40
4
x
x
11
Discard 4. The number is 7.
Chapter 10 Radicals, Radical Functions, and Radical Exponents
49.
16 2
fx x x
To find the x-intercepts, set the function equal to 0
and solve for x.
Check
9x
:
916 9 2
25 9 2
532
0
The only x-intercept is 9.
50.
23 21fx x x
To find the x-intercepts, set the function equal to 0
and solve for x.
02321
xx
Check
2x
:
22 3 22 1
51.
2
d
t
2
2
1.7
1.7
2.9
d
d
d
The vertical distance was about 2.9 feet.
53. It is represented by the point (5.4, 1.16).
54. It is represented by the point (2.9, 0.85).
55. a.
() 4.4 38
fx x
(25) 4.4 25 38 16
f
16% of Americans earning $25 thousand
annually report fair or poor health. This
22
24
4.4
30
x
x
Section 10.6 Radical Equations
56. a.
() 4.4 38
fx x
(60) 4.4 60 38 3.9
f
graph by 0.1%.
b.
() 4.4 38
fx x
24 4.4 38
x
57.
1/3
1/3
1/3
87 29
87 29
29 29
3
x
x
x
58.
1/3
1/3
29
58 29
fx x
x
59.
3/2
365 0.2
x
3/2
3
33
3
365 0.2
3,330,625
3,330,625
x
x
x
3
33
3
193,600
193,600
58
x
x
x
The average distance of Mercury from the sun is
approximately 58 million kilometers.
The solution set is
4.
Chapter 10 Radicals, Radical Functions, and Radical Exponents
70.
2
31
xx
71.
43
xx
72.
42
x
73. does not make sense; Explanations will vary.
Sample explanation: You should always substitute
74. makes sense
75. does not make sense; Explanations will vary.
77. false; Changes to make the statement true will vary.
A sample change is: The first step is to square both
sides, obtaining
2
644
xxx
.
78. false; Changes to make the statement true will vary.
80. false; Changes to make the statement true will vary.
A sample change is: Note that if x is negative, then
–x is positive. Thus an equation of this form can
have a real solution. In this case the solution is
1.
3
x
2
3
729
531441
531441
81
xx
xx
x
x
22
93
xx
x
98118xx
98118
16
x
x
Section 10.6 Radical Equations
Check:
93
16 16 9 3
xx
83.
2/3
3/2
2/3 3/2
425
425
x
x
Check:
2/3
129 4 25
84. 34 3 2 1 1
12 45 141 426
4 15 47 142 425
85.
2
2
2
312618
4
28
312 4
xx
x
xx
xx
32 1
163
x
x
32 2
xx
35 35
yxyx
Chapter 10 Radicals, Radical Functions, and Radical Exponents
88.
2
2
(7 3 )( 2 5 ) (7)( 2) (7)( 5 ) ( 3 )( 2) ( 3 )( 5 )
14 35 6 15
15 29 14
xx xx xx
xx x
xx
89.
742 742252
252 252252
742252
252252
10.7 Check Points
1. a.
64 64 1 64 1 8i
b.
11 11 1 11 1 11i
2. a.
(5 2 ) (3 3 ) 5 2 3 3
8
iiii
i
Section 10.7 Complex Numbers
3. a.
2
7(29) 7279
14 63
14 63( 1)
63 14
iii ii
ii
i
i
b.
2
(5 4 )(6 7 ) 30 35 24 28
i i iii
5.
62 62 43
43 4343
iii
iii
2
22
24 18 8 6
4(3)
iii
i
6.
32 32 4
444
iii
iii
2
2
12 8
16
ii
i
c.
35 34 2 17 17
() (1) (1)iiiii i ii
10.7 Concept and Vocabulary Check
1.
1
; 1
8.
25i
9.
4i
10.
1
; 1
2.
49 49 1 7i
3.
23 23 1 23 1 23i
4.
21 21 1 21i
Chapter 10 Radicals, Radical Functions, and Radical Exponents
9.
108 36 3 1
36 3 1
63i
12.
7 47 4172
i
13.
15 3 15 3 1
15 3 1
15 3
i
16.
3273931333
i
17.
32 5
32 5 352
35 21 83
ii
ii ii
ii
20.
26 4 26 4 25ii iii
23.
94 103
9 4 10 3 9 10 4 3
910 43
ii
ii ii
i
35 27 29
ii
26.
75 911 75 911
16 16
iiii
i
27.
5 4 13 11
ii
29.
8149
8149 1489
14 8 9 14 17
ii
ii ii
ii
30.
15 12 11 15 12 11 12 26
iiii i
32.
45865
ii
Section 10.7 Complex Numbers
35.
2
37 5
3 7 3 5 21 15
21 1 15 21 15
ii
ii i i i
ii
38.
2
635 1830 18301
30 18
ii ii i
i
39.
2
3 45 121545
12 15 4 5 1
12 5 15 4
719
ii iii
ii
ii
i
42.
2
8 4 3 2 24 16 12 8
24 28 8 1 16 28
ii iii
ii
43.
2
63 25
12 30 6 15
12 30 6 15 1
12 15 30 6 3 36
ii
ii i
ii
ii i
46.
27 27 4491 449
53 53 0
ii
i
49.
3232
932
ii
i
32
i
2
2
921 92
11 11 0
i
i
50.
2
5353253
25 3 1
ii i
52.
22
32 9232 4
912 4 1 512
iii
ii
53.
22
52 25210 4
25 20 4 1
25 4 20 21 20
iii
i
ii
54.
22
5 3 25 30 9
iii
Chapter 10 Radicals, Radical Functions, and Radical Exponents
57.
2
94
9141326
61 6 60
ii i
i
60.
2
3363 663
63 1 63 63 0
ii i
i
62.
2
9535 35
35 1 35 35 0
ii i
i
64.
334 123123
444161161
12 3 12 3
17 17 17
iii
iii
ii
66.
2
22
10 5 1
552105
222 41
2
10 5 10 5 5 10 12
41 5 5 5
i
iiiii
iii i
ii ii
28 21
25 25
i
68.
2
9 9 12 918 918
ii i
2
18 12 1 12 18
941
94
12 18 12 18
ii
i
ii
10 15 10 15 15 10
4 9 13 13 13
ii i
71.
2
22
11112
111 1
iii ii
iii i
11 2
Section 10.7 Complex Numbers
73.
23 233
333
iii
iii
74.
2
22
23 233 6293
333 3
611 3 1 611 3
91 91
311 3 11
10 10 10
iiiiii
iii i
ii
ii
75.
52 52 32
32 32 32
iii
iii
76.
2
2
2
63 63 42 2412 12 6
42 4242 42
iii iii
iii i
77.
45 4537
37 37 37
iii
iii
949 58
23 43
58 58
i
78.
2
2
2
2
5 5 3 2 15 10 3 2
32 32 32 32
15 7 2 1 15 7 2
941
94
17 7 17 7 17 7
9 4 13 13 13
iii iii
iii i
ii
i
ii i
81.
2
2
85 85 2 16 10
222 4
16 10 1 10 16
iiiii
iii i
ii
Chapter 10 Radicals, Radical Functions, and Radical Exponents
83.
2
2
47 473 12 21
3339
iiiii
iii i
84.
2
2
51
55 5
44 41
4
51 15
444
i
iiiii
iiii
ii
87.
55
11 2
1iii ii
88.
77
15 2
1iii ii
91.
100 100
200 2
11ii
92.
200 200
400 2
11ii
93.
88
17 2
1iii ii
97.
4
99
92
11
iiii
6
11 1
iii
99.
12
24 2 2
12
1
11
ii i
101.
222
22
2
231 3 3
22 3 3 3
25 3 9
ii ii
iii i
ii i
222
22
2
16 8 18 9 1
16 10 9 1
15 10 8
15 10 8 1
23 10
iii i
ii i
ii
i
i
22
Section 10.7 Complex Numbers
105.
516381
106.
58318
54213921
52 2 33 2
10 2 9 2
19 2 or 0 19 2
ii
ii
ii
108.
210
44
840
43 42
ii
ii
ii iii
210
22
11 11
11 1
221
111
22 22
11
1
22 1
2
ii
i
iii
ii
i
ii
110.
2
25
fx x x
44
0
111.
3
fx x i
;
42
gx x i
113
142
fi
gi
11342
fg i i
810 3 1
11 10
i
i
113.
2
19
2
x
fx x
22
319
919
323 23
9119 10
23 23
10 2 3
2323
ii
fi ii
ii
i
ii
Chapter 10 Radicals, Radical Functions, and Radical Exponents
114.
211
3
x
fx x
15 20
25 25
34
55
i
i
115.
45 37
E
IR i i
116.
2
610 9 15 6 15 1
61521
ii i i
ii
The voltage of the circuit is
21 i
volts.
117. Sum:
515515
ii
118. – 130. Answers will vary.
131.
916916
34 7
ii
ii i
132.
2
2
999
33 9 9ii i
137. false; Changes to make the statement true will vary.
A sample change is: All irrational numbers are
complex numbers.
138. false; Changes to make the statement true will vary.
A sample change is:
2
22
37 37
ii
2
2
2
2
73 7353
53 53 53
35 21 15 9
53
35 21 15 9 1
25 9
iii
iii
iii
i
ii
i
140. true