Section 10.5 Multiplying with More Than One Term and Rationalizing Denominators
19.
2735
23 25 73 75
6102135
22.
35 23 45 53
12 5 15 15 8 15 10 3
60 7 15 30 30 7 15
24.
2
27 221479214
25.
2
22
3
323
323
xy
xxyy
xxyy
28.
62 62 642
29.
253253
6
32.
43 32 43 32
16 3 12 6
12 69 2 48 18 30
65
65
xx
xx
34.
43 127xx xx
35.
33
33 3 3
45
5445
xx
xx x x
Chapter 10 Radicals, Radical Functions, and Radical Exponents
38.
33236
55 55
235
55
23
55
22
2
2
xy xy xxy y
xxy yy
xxy yy
41.
2
11 11 11
11 11
x
xxxx
xx
x
x
44.
3123
12 12
33333
43
yy
yyyyy
3
3
y43y
y
y
48.
33
33
23
10 10 5 1025 1025
5
55
55
5
3
225
5
3
225
33
22
33
50.
3
33
33
23
33
44
44
44
48 8 6 2
44
3
6
2
3
6
2
2
33
22
44 4
xx
33
3
3
22
yy
yy
y
y
33
33 3
55
yyy
Section 10.5 Multiplying with More Than One Term and Rationalizing Denominators
55.
33
22
33333
222 22
33
333
772 72
22222
74 74
2
2
xx
xxxxx
xx
x
x
x
57.
2
3
33
32222
333
22
333
22 33
33 3
2
3
222
22
2
xy
xy xy xy x y
xy xy
xy x y x y
xy
xy
59.
44
33
44
44
33
33 3
xx
xx
xxx
44
33
44
33
xx
x
x
62.
55
33
5555
2423 55
55
33
10 10 2 10 2
1622 2
10 2 5 2
2
xx
xxx x
xx
xx
2
x y
3
5
8
xxy
64.
242
22 2
5
33333242
55 55
242 242
55
555
5
2
33 3
82 22
34 34
2
2
xy
xy xy xy
xy xy xy x y
xy xy xy xy
xy
xy
2
3
9
33
93 93
3
3
3
y
xy y
yy
xy
xy
33
3
y
33
y
xy
xy
Chapter 10 Radicals, Radical Functions, and Radical Exponents
68.
332
554
22
2
22
3
150 150 25 6
56 56
56 56
56
aaaa
bbbb
aa aa b
bb bb b
aab aab
bb
bb
aab
b
70.
46 2 2
34
2
5
3
15 3
33 3
3
33 (3)
mn mn mn
mn
nm nm nm
73.
58 36 22
33
12 12
88
xy xy xy
222
3
3
3
222
3
33
2
233
3
12
2
12
2
12 12
2
xy x y
xy
xy
xy x y
xy xy
xy xy
411 39 2
33
27 27
xy xy xy
32
3
2
3
32 2
33
15
3
15
3
xy xy
xy
xy xy x y
=
−
=⋅
−
Section 10.5 Multiplying with More Than One Term and Rationalizing Denominators
76.
15 6 1
15 15 6 1
61
61 61 61
15 6 1
361 363
5
77.
13 13 11 3
11 3 11 3 11 3
2
78.
17 17 10 2
10 2 10 2 10 2
515
65 63
53
65 63
2
2
35 33
2
abab
aa ab
Chapter 10 Radicals, Radical Functions, and Radical Exponents
84.
35 35 5 2 3 5
52 35 52 35 52 35
35 5 2 3 5 35 5 2 3 5
25 2 9 5 50 45
35 5 2 3 5
75 2 3 5
5
35 2 21 5
53
2
86.
11 5 11 5 11 5
11 5 11 5 11 5
11 2 55 5 16 2 55
11 5 6
28 55 855
63
Section 10.5 Multiplying with More Than One Term and Rationalizing Denominators
90.
26 5 26 536 5
36 5 36 5 36 5
36 530 5 41 530
96 5 54 5
41 5 30
92.
33 3
333
69
9
xy xyy x
yxyxyx
xy x y
yx
93.
333333
223 23 6
94.
5555
335 15
Chapter 10 Radicals, Radical Functions, and Radical Exponents
100.
2
ab abab
ab abab
ab
aabb
102.
777
77
7
77
77
xxxxxx
xx
xx
xx
7
7
1
7
xx
xx
104.
22 22
xy xyxy
xy xy x y
Section 10.5 Multiplying with More Than One Term and Rationalizing Denominators
106.
115
55
555
555 5
5555
55 5 65
55
108.
43
44
44
43
4
4
44 4
20 20 2
88
22
2
20 8
82
8108 98
15 15 5 15 3 15
15 15 15
15 15 5 15 3 15
15
13 15
15
22 23 53 52
23
22 23 53 52
1
23 22 53 52
Chapter 10 Radicals, Radical Functions, and Radical Exponents
113.
2
64fx x x
2
3 13 3 13 6 3 13 4
96
0
13 13 18 6 13 4
f
115.
9fx x
2
2
35 35 9 35 9 35
935935
935
81 9 5
81 45
36
6
ff
117.
2
51
w
h
251
51 51
118.
7223 723
66
72
3
2
73
3
3
The perimeter is
82
inches.
2
Area 8 1 8 1
88
lw
81
81 7
The area is 7 square inches.
120.
Perimeter 4 4 2 3 2
83 42
s
The perimeter is
83 42
inches.
129.
111xxx
Section 10.5 Multiplying with More Than One Term and Rationalizing Denominators
130.
2
2 2 4 for 0xxx x
131.
2
11xx
132.
33
3xx
xx
133. makes sense
134. does not make sense; Explanations will vary.
Sample explanation:
2
()ab
does not equal
22
.ab
Actually,
22 2
() 2 .ab a abb
Thus,
22 2
(2 5) (2) 2 2 5 (5)
22105
A sample change is:
2
22
37 37 32
32 32 32
3237314
32
139. true
140. false; Changes to make the statement true will vary.
A sample change is:
22 2
7277
14 49
xxx
xx
7423
745
45
7
x
x
x
Chapter 10 Radicals, Radical Functions, and Radical Exponents
142.
2
23 23
143.
1
234
232
1
232232
22
232 232
22634
23 2
144.
2
23
24
xx
23
222
xxx
145.
342and455
36 40
xx
xx
42 41
52 4
xx
xx
148.
2
2
2
416164(4)
41616416
4200
4( 5) 0
xx x
xx x
xx
xx
Section 10.6 Radical Equations
10.6 Check Points
1.
22
348
34 8
x
x
2. 17 2
15
x
x
A principal square root cannot be negative. Thus,
this equation has no solution.
The solution set is
.
3.
67 2
67 2
xx
xx
67 2
6(3) 7 (3) 2
25 3 2
53 2
22
xx
The solution set is
1, 3 .
4. 532xx
22
532
532
4
xx
xx
x
Check:
532
45 43 2
912
31 2
22
xx
The solution set is
4.
3
(27) 3 0
27 3 0
33 0
00
The solution set is
12 .
Chapter 10 Radicals, Radical Functions, and Radical Exponents
6.
() 3.5 38
73 3.5 38
fx x
x
10.6 Concept and Vocabulary Check
1. radical
2. extraneous
3.
21x
;
2
14 49xx
10.6 Exercise Set
1.
22
324
32 4
3216
x
x
x
2.
22
518
51 8
x
x
585
17
x
x
Check:
517 4 9 0
85 4 9 0
81 9 0
32 5
3225
327
9
x
x
x
x
The solution set is
9.
Section 10.6 Radical Equations
7.
2
2
2
2
78
78
78
780
810
xx
xx
xx
xx
xx
8.
2
2
2
67
67
67
xx
xx
xx
9.
22
51 1
51 1
xx
xx
10.
22
2
2
21 7
21 7
2 1 14 49
01648
0124
xx
xx
xxx
xx
xx
430
310
xx
xx
Apply the zero product principle.
30 or 10
31
xx
xx
2
320
210
xx
xx
20 or 10
xx
Chapter 10 Radicals, Radical Functions, and Radical Exponents
13.
233
32 3
xx
xx
14.
2
2
3375
35 37
35 37
xx
xx
xx
15.
25 4
xx
16.
22
62 53
62 53
xx
xx
17.
3
33
3
2113
x
The solution checks. The solution set is
5.
19.
3
3
33
2640
264
x
x
43125
4128
32
x
x
x
Section 10.6 Radical Equations
21.
22
77
77
xx
xx
x
74914xx
22. 82xx
22
82xx
x
8x
44
84 4
x
x
23.
22
213
23 1
23 1
xx
xx
xx
x296 1xx
1
24.
444xx
22
44 4
44 4
xx
xx
14
5
x
x
The solution checks. The solution set is
5.
25.
24 1 9 5xx
60
6
x
x
or
20
2
x
x
Check
6x
:
246 1 9 6 5
Chapter 10 Radicals, Radical Functions, and Radical Exponents
Check
2x
:
242 1 9 2 5
26.
234 1xx
22
23 3
23 3
xx
xx
Check
3x
:
23 3 4 3 1
20 4 4
04 4
44
Check 7
x
:
27 3 4 7 1
27.
1/3
1/3
23 46
23 2
x
x
28.
1/3
1/3
36 58
36 3
x
x
1/4
4
1/4 4
31 2
31 2
3116
315
5
x
x
x
x
x
The solution checks. The solution set is
5.
30.
1/4
23 710
x
31.
1/2
1/2
284
24
x
x
Section 10.6 Radical Equations
32.
1/2
386
x
33.
23 21xx
22
23 21
23 21
xx
xx
34.
2371xx
22
21 3 7
2137
xx
xx
35.
1/3
1/3 2
317
010
010
xxx
xx
xx
0100
10
xx
x
Both solutions check. The solution set is
2, 4 .
37.
1/4 1/4
44
1/4 1/4
82
82
xx
xx
Chapter 10 Radicals, Radical Functions, and Radical Exponents
39.
5fx x x
75
75
xx
xx
Check
11x
:
11 11 5 11 16
15 7
40.
2fx x x
22
2
2
42
24
24
2816
0918
xx
xx
xx
xxx
xx
41.
1/3
516fx x ;
1/3
12gx x
1/3 1/3
516 12
xx
42.
1/4
92fx x ;
1/4
518gx x
1/4 1/ 4
92 518
4
xx
x
Check:
1/4 1/4
1/4 1/ 4
1/4 1/4
94 2 54 18
36 2 20 18
38 38
The solution is 4.
43.
2
3
V
rh
π