Chapter 9 Radical Expressions, Functions, and Equations
172
112.
44
44
xyx y
xyxy
++
⋅
−+
114.
()
33
22
322
34
33
3
23
() 49 12
49 12
4108
427 4
43 4
12 4
fx gx x x x
xx x
xx
xxx
xx x
xx
⋅= ⋅
=⋅
=
=⋅
=⋅
=
)
)
)
)
116.
(
)
(
)
(
)
(
)
()
2
24
428
68
fxgx x x
xxx
xx
⋅=+ +
=+++
=+ +
118.
(
)
2
32 5 10 32 10 510
10 10 10
6552
10
rrrr r
rr r
rr
r
−−
⋅=
−
=
120.
(
)
() ()
33 3
33 33
33
33
33
9418 715
49187915
4162 7135
4276 7275
43 6 73 5
−−
=− −−
=− +
=− ⋅ + ⋅
=− ⋅ + ⋅
()
3
5
95
5
n
mn
n
=
126.
() ()
1
27 14
fxgx x x x
⋅= ⋅
Section 9.5 Multiplication and Division of Radical Expressions
173
126. (continued)
()
()
27
114
fx xx
gx x
=
128. a. Area of a circle = 2
rπ.
Radius is half the diagonal. 2
2
s
r=
b.
(
)
22
16
4
4; 8
22 2
s
sA
π
ππ
π==== =
The area is 2
8 in.π
132.
(
)
32
3
3
32
3
3
333
23
VPP
VPP
PPP
−
−⋅=
Mindstretchers
1.
(
)
(
)
?
2
322 6322 10
−−−+=
2. a.
()
22
yyxxyxxy
xxxx
yyy
yyyy
yy
x
=⋅= ⋅= =
11 11
11 11
11 11
11 11
1
xx
xx xx
xx
xx xx
x
⎛⎞ ⎛⎞
⎛⎞ ⎛⎞
+−
⎟⎟
⎜⎜
⎟⎟
⎜⎜
⎟⎟
+
⎜⎜
⎟⎟
⎜⎜
⎟⎟
⎟⎟
⎜⎜
⎟⎟
⎜⎜
⎟⎟
⎟⎟
⎜⎜
⎝⎠ ⎝⎠
−+ +−
⎝⎠ ⎝⎠
=⎛⎞ ⎛⎞
⎛⎞ ⎛⎞
−+
⎟⎟
⎜⎜
⎟⎟
⎜⎜
⎟⎟
−
⎜⎜
⎟⎟
⎜⎜
⎟⎟
⎟⎟
⎜⎜
⎟⎟
⎜⎜
⎟⎟
⎟⎟
⎜⎜
⎝⎠ ⎝⎠
+− −+
⎝⎠ ⎝⎠
−
=
11
11 11
11 1
2
xx
xx xx
xx x
−−
=
−+ −−−
−
−− −
Chapter 9 Radical Expressions, Functions, and Equations
174
3. a. 22yx x=+⋅−
b. The domain of the expression 2x+ is
2,x≥− and the domain of the expression
2x− is 2.x≥ Since the expression 2x−
cannot be evaluated for any 2,x< the domain of
9.6 Solving Radical
Equations
Exercises
716
23
x
x
−=
=
6.
22
415
x
+=
10. 10 1
9
x
x
+=
=−
Since 0,x≥ there is no solution.
12.
87
15
x
x
−=
=
Section 9.6 Solving Radical Equations
175
16.
()
22
311 2
39
39
381
27
n
n
n
n
n
−=−
=
=
=
=
20. 21 3 15 24
3153
315 3
x
x
x
−−=
−−=
−=−
Since 3150,x−≥ there is no solution.
24.
()()
()
22
6113 7
611 3 7
6119 7
611963
xx
xx
xx
xx
−= −
−= −
−= −
−= −
28.
(
)
(
)
(
)
(
)
22
2
2
2
823
823
3100
520
nn
nn
nn
nn
−= −
−=−
+−=
+−=
50 20
nn
+= −=
() ()
?
2
28232
48 4
−= −
−=−
4 is not a real number,
so 2 is not a solutionn
−
=
(
)
(
)
2
2
10 9 8
9180
630
pp
pp
pp
+=−−
++=
++=
60 30
63
pp
pp
+= +=
=− =−
Chapter 9 Radical Expressions, Functions, and Equations
176
30. (continued)
Check 3.p=−
32.
()
()
22
2
32
32
32
yy
yy
yy
−=
−=
−=
34.
()
()
22
3610
613
61 3
aa
aa
aa
−+ −=
−=
−=
34. (continued)
1
Check .
a=
00
=
36.
22
25 1
xx
+=+
11
≠−
2 is not a solution.−
Check 2.x=
()
?
22 5 2 1
+=+
Section 9.6 Solving Radical Equations
177
38.
5932
5923
xx
xx
++=
+= −
Check 0.x=
() ()
()
?
?
?
?
50 9 3 20
09320
930
330
60
++=
++=
+=
+=
≠
0 is not a solution.x=
40.
(
)
(
)
123
13 2
2500
xx
xx
xx
−+ =
−=−
−−=
20 500
250
xx
xx
−= − =
==
Check 2.x=
()
?
?
?
21 22 3
143
123
−+ =
+=
+=
Chapter 9 Radical Expressions, Functions, and Equations
178
42.
(
)
(
)
22
46 2
46 2
rr
rr
−+=−−
−+ =−−
168662
rrr
−+++=−
44.
()()
()
22
14 3 3
14 3 3
14 3 6 3 9
22 6 3
21 6 3
aa
aa
aa a
aa
aa
−= ++
−= ++
−=++ ++
−= +
−= +
Check 13.a=
?
14 13 13 3 3
−= ++
44. (continued)
Check 2.a=−
()
?
14 2 2 3 3
−− = − + +
(
)
(
)
2
650
150
xx
xx
−+=
−−=
10 50
15
xx
xx
−= −=
==
Check 1.x=
() ()
?
21 1 1 31 1
−+= +
?
?
91 16
314
+=
+=
Section 9.6 Solving Radical Equations
179
48.
()
()
33
33
33
323
82
82
88126
xx
xx
xxxx
−=−
−=−
−=− + −
50.
()
()
22
7423
7432
74 32
xx
xx
xx
+−=
+= +
+=+
52.
()
()
3
3
3
33
3
13 9
913
4
4
64
p
p
p
p
p
+=
=−
=−
=−
=−
54. (continued)
() ()
?
2
?
Check 6.
611736
a=−
−−=−−
56.
()
22
22
2
2
18 2 8
18 2 64
dxh
h
h
=+
=+
=⋅+
58. a.
()
(
)
2
2
64
64
64
vd
vd
v
d
=
=
=
b.
2
80 6400 100
64 64
d== =
100 ft
Chapter 9 Radical Expressions, Functions, and Equations
180
Mindstretchers
1. Equivalent equations have the same
solutions. Raising each side of the radical
2. a. The graphs intersect at the point
(
)
2,1 .
b.
()
()
(
)
(
)
22
2
2
13
13
196
7100
520
xx
xx
xxx
xx
xx
−=−
−=−
−= − +
−+=
−−=
c. They are the same.
d. A grapher can be used to solve a radical
equation by entering the expressions on
3. a. 292x−=
(
)
(
)
2
22
92
550
x
xx
⎛⎞
⎟
⎜−=
⎟
⎜⎟
⎜
⎝⎠
+−=
50 50
55
xx
xx
+= −=
=− =
Check 5.x=−
()
?
2
?
?
592
25 9 2
−−=
−=
42
22
=
=
The solutions are 5− and 5.
Section 9.7 Complex Numbers
181
3. b.
()
()
2
3
2
22
3
21 2
21 2
x
x
+=
⎛⎞
⎟
⎜⎟
+=
⎜⎟
⎜⎟
⎜
⎝⎠
9
Check .
2
x=−
()
()
?
2
3
?
3
?
91 2
82
64 2
42
22
−+ =
−=
=
=
=
7
Check .
2
x=
9.7 Complex Numbers
Exercises
64 64 8 8
12. 10 10i−=
16. 72 72 36 2 6 2ii i−= =⋅ ⋅ =
18. 21 2i−−=−
20. 773 7
33 7
36 6 2
36
i
ii−= = =
22. 11
32 32
22
116 2
i
−−=−
Chapter 9 Radical Expressions, Functions, and Equations
182
28.
(
)
(
)
()( )
10 4 9 4 10 4 9 4
10 9 4 4
1
ii ii
ii
−−−=−−+
=−+−+
=
)
)
34.
(
)
(
)
)
)
42 4 124 1
+−− −−
36.
(
)
(
)
2
91634
12
12
ii
i
⋅− =
=
=−
)
)
)
)
)
40.
()
()
2
11 5 55
55 1
55
ii i⋅=
=−
=−
42.
()
()
2
68 48
48 1
48
ii i−=−
=− −
=
)
48.
(
)
(
)
2
12 49 2 7
27
27
ii
ii
i
−+−=+
=+
=−
52.
(
)
(
)
3514
F O I L
ii
++
2
F O I L
816 3 6
813 6
iii
i
=− + −
=− +
774
i
=− +
58.
(
)
(
)
2
36 63
F O I L
18 9 36 18
18 27 18
36 27
ii
iii
i
i
−− −
=− + − +
=− − −
=− −
60.
(
)
(
)
(
)
(
)
22
737373
ii i
+−=−
Section 9.7 Complex Numbers
183
64.
(
)
(
)
(
)
(
)
(
)
22 2
2
34 3 234 4
924 16
iii
ii
−=− +
=− +
68.
() ( )
22 2
Complex Conjugate Product
Number
52 52 5 2 254
25 4
29
ii ii+− −=−
=+
=
70.
()
2
22
Complex Conjugate Product
Number
43 43 4 3 169
16 9
25
iiii−+−=−
=+
=
72.
Complex Conjugate Product
Number
74.
(
)
(
)
(
)
22
Complex Conjugate Product
Number
05 05 5 25 25iiii+−−=−=
76.
Complex Conjugate Product
78.
1123
23 2323
i
iii
−−−
=⋅
++−
()
()()
2
333
33
33
93
9
93
10
93
10 10
iii
i
ii
i
i
i
i
−−+
+
=−+
+
=−
+
=
=+
82.
2
992
222
18
4
18
4
i
iii
i
i
i
−
=⋅
−
−
=−
−
=
2
25
15
25
3
5
i
i
i
=−
=
=
Chapter 9 Radical Expressions, Functions, and Equations
184
88.
24 243
333
iii
iii
−−+
=⋅
−−+
90.
45 4553
53 5353
iii
iii
−−−
=⋅
++−
92.
696342
4242
44
ii
ii
+− + −
=⋅
+−
+−
94. 9993
39
381
i
i
+− +
=−
−−
(
)
()
33
31 3
i
i
+
=−
96.
(
)
5
20 4 5
11ii===
()
71
i= = ⋅= ⋅= ⋅=
104.
(
)
()
3
11 3 14 12 2 4 2 11 1ii i ii i i⋅= = ⋅= ⋅=⋅−=−
41
42
5
i
=+
−−
=
21 66 48
27 66
i
i
=− + +
=+
114.
()()
8593
VIZ
ii
=
=+ +
Section 9.7 Complex Numbers
185
Mindstretchers
1.
(
)
43 4 3 4 3 1
n
nn n
iiiii ii
+=⋅= ⋅=⋅−=−
2. Answers may vary.