Section 10.3 Multiplying and Simplifying Radical Expressions
10.3 Exercise Set
1. 35 35 15
2. 75 35
3.
333 3
29 29 18
9.
555
334
5
646424xxxx x
10.
44
23
4
6318xx x
11.
2
33 33 9xx xx x
12.
2
66 36xx x
16. 2525
52 52
xx
x
17.
444 4
333
xxx
19.
232232
77 7
23 2
7
53
7
711 711
711
77
xy xy xy xy
xxyy
xy
20.
23 34 2334
999
12 3 12 3
xy xy xxyy
25. 75 25 3 25 3 5 3xx xx
26. 40 4 10 2 10xxx
27.
33
33 3
16 8 2 8 2 2 2
28.
3
33
54 27 2 3 2
29.
33 3
33 3
3
27 27 27 3xx xx
33.
2
36 2 6 2fx x x
34.
2
81 2 9 2fx x x
Chapter 10 Radicals, Radical Functions, and Radical Exponents
36.
33
33
3
48 2 8 6 2
226
fx x x
x
39.
76 6 3
x xx x xxx
40.
54 2
xxxxx
41.
89 88 88
44
xy xyy xy y
xy y
45.
86262
33 33
22
3
yyyyy
yy
46.
11 9 2 3 2
33 3
yyyyy
50.
917 9152 35 2
33 3
32 8 4 2 4xy xy y xy y
532
33
53.
17 15 2 15 2
55 55
32
5
yyyyy
yy
54.
18 15 3 3 3
55 5
yyyyy
55.
617 5 152
55
515 2
55
64 32 2
32 2
xy xxy y
xy xy
4
22
25
xx
58.
44 4
11 8 3 2 3
96 16 6 2 6
xxxxx
59.
10 8 2
44
333
xxx
Section 10.3 Multiplying and Simplifying Radical Expressions
61.
12 2 12 2 24
46 4 6
26
65.
2
12 3 12 3
36 6
xx xx
xx
68.
223 22
510 50 252
52
xy xy x y x y y
xy y
69.
2 5 3 40 2 3 5 40 6 200
6 100 2 6 100 2
610 2 60 2
74.
7 4 11 10 5
212 24 46 26xx x xxxx
75.
42 12
33
25 5
xy xy
76.
7 4 12 11 13
33 3
9212
3
69 54
27 2
xy xy x y
xxy y
2
xyz x z
78.
233 4 6 649
444
4832xyz xyz xyz
4
4248 2 2
4
16 2 2 2xxyzz xyz xz
79.
462 74
55
462 74
5
88
xyz xyz
Chapter 10 Radicals, Radical Functions, and Radical Exponents
81.
7
3
3
7
3
8
3
xy xy
xy xy
xy
83.
2372
3
2362
3
2362
33
22 2
3
33 2
3
254
2272
227 2
23 2
62
xy xyz
xy xyyz
xy xy yz
x y xy yz
xy yz
85.
36 35
55
53
55
364 3322
332 2
y xy y xyy
yy xy
86.
2 7 11 17
5
2 7 10 15 2
5
2 7 10 15 2
55
432
432
432
xy x y
xy x xy y
xy x y xy
33
62
xy
88.
23 4
52 10
xyz xyz xz xz
632
632 2 2
632 22
632 22
5210
520
545
545
x y z xyz xz
xyz xyz
xyz xz y
xyz xz y
4
33 3
4
34 3
62
12
xy y x
xy x
Section 10.3 Multiplying and Simplifying Radical Expressions
91.
3
2
x
dx
372
72 2
d
93.
42Wx x
6426412
W
94.
42Wx x
10 4 2 10 4 20
W
95. a.
44
44
7.644 7.644
32 32 16 2
7.644 3.822
22 2
C
The cardiac index of a 32-year-old is
4
3.822
2
.
96. a.
4
4
7.644
80 80
7.644
16 5
C
minute per square meter. This is represented on the
graph as the point (80, 2.56) .
97. – 102. Answers will vary.
The graphs do not coincide. Correct the
simplification.
22
84222
xxx
.
Chapter 10 Radicals, Radical Functions, and Radical Exponents
105.
2
363 13
xx x
106.
32
3
24 4
xx x
107. makes sense
108. makes sense
111. false; Changes to make the statement true will vary.
A sample change is:
2 5 6 5 12 5 5 12 5 60
115. If a number is tripled, its square root is multiplied by
3.
For example, consider the number 4 and its
triple, 12.
4 2 12 4 3 2 3
Thus, if a number is tripled, its square root is
3
1
1
3
1
22
2
2
2
2
xxgx
xgx
x
xgx
x
2
2
2
2
1
111100
222111
xfxx
f
f
Section 10.4 Adding, Subtracting, and Dividing Radical Expressions
119.
2121and2212
222 210
xx
xx
120. Multiply the first equation by –3, the second equation by 2 and solve by addition.
15 6 6
8 6 8
7 14
2
xy
xy
x
x
121.
32
64 27 4 3 16 12 9xxxx
122. a.
21 10 31xx x
b.
21 2 10 2 31 2
10.4 Check Points
1. a.
8 13 2 13 10 13
Chapter 10 Radicals, Radical Functions, and Radical Exponents
2. a.
320 545 345 595
32 5 53 5
65 155
21 5
3. a.
333 3
33
33
3
324 581 383 5273
32 3 53 3
63 153
93
4. a.
33 333
3
33 3
24 24 8 3 8 3 2 3
125 5
125 125 125
5. a.
55
44 4 2
40 40 20 4 5 4 5 2 5
2
2
xx
xx x x
x
x
10.4 Concept and Vocabulary Check
Section 10.4 Adding, Subtracting, and Dividing Radical Expressions
1. 5; 8;
13 3
2. 9; 4; 3; 2;
3
10.4 Exercise Set
1.
85 35 8 3 5 115
2.
73 23 93
7.
313 2 5 213 4 5
313 213 25 45
32 13 24 5
13 2 5
12.
520 545 52535
13.
712 75 743 253
15.
2735
23 25 73 75
6102135
16.
545 220 595 245
53 5 22 5
xx x x
xx
63 33 93
19.
32
345 5 395 5
33 5 5
95 5
915
xx xxx
xx x
xx x
xx
Chapter 10 Radicals, Radical Functions, and Radical Exponents
22.
33
33
33
24 81 8 3 27 3xy y x xy y x
33 3
23 33 53yx yx yx
25.
918 2
92 2
xx
xx
32 2
31 2
42
xx
x
x
28.
42 2 3 2 2
333 3
22
33
2
3
454 5
45
9
xy xxy xxy xxy
xxy xxy
xxy
29.
11 11 11
32.
3
3
11 11
64 4
3
25 5
25
22
5
yy
y
xx
y
36.
32
84 4
50 25 2 5 2
81 9 9
xxxxx
yy y
33
40.
33
86222
315 5 5
81 27 3 3 3
82 2
xxxxx
yy y
41.
642 2
644 4
4822
48
99 9
9
yyyyy
y
xxx
x
y
Section 10.4 Adding, Subtracting, and Dividing Radical Expressions
44.
55
14 10 4 2 4
515 3 3
64 32 2 2 2
xxxxx
yy y
45.
40 40 84222
5
5
50.
3
2
72 36 6
2
xxx
x
51.
53 53
42 2
xy xy xy xy
xy
xy
52.
76
44 22
32
xy xy xy
xy
56.
711 711
11 13
42
42
10 12 5 6
54 54 18
3
3
92 3 2
ab ab ab
ab
ab
aabb ab ab
72 172 136
xy xy xy
60.
53
33
23 2
3
33
250 125 5
2
xy xy y x
x
61.
10 8 10 8
482
8
4
422
22
4
810 88 2
44
22 2 22
4
32 32 16
2
2
16 16
2 or 2
xy xy xy
xy
xy
xy xyy
xy y xy y
Chapter 10 Radicals, Radical Functions, and Radical Exponents
66.
333 33
3
3
ab ab
ab
ab
67.
32 18 16 2 9 2
57 5 7
16 2 9 2
57
68.
27 75 9 3 25 3
69.
2332
2222
38 532 18
34 2 516 2 9 2
xxy y x xy
xy x y x x xy x
2
2
63 1225 3
63373
xy x x xy
xy x x xy
3
2
52 10 8
521042
52 1022
xxx
xx x x
xx x x
72 852
72 402
xx x x
xx xx
Section 10.4 Adding, Subtracting, and Dividing Radical Expressions
73.
2
2
81
275
3
xy
xxy
xy
74.
2
23
2
964
58
32
xy
xy
xy
75.
432 3
33
3
32 1
3
15 80 75 5
52 25
xxy xy
xxy x
32 3
3
321
4
3
3
3
3
3
80 5
33
2
340 35
385 3 5
xy xy
xxy x
xxy xy
xxyxxy
76.
432 3
33
3
32 1
3
16 48 20 2
83 4
xxy xy
xxy x
77.
2
2
32
48 48
3
3
16 16 4
fxx
x
gx
x
xxxxx
2
Therefore, we need 5x.
Domain:
5,
79.
366
34
32
32
333
32 32 16
2
2
82 22
fxx
xx
gx
x
xx xx
Our only restriction here is that we cannot divide by
0. Thus, we need 0x.
Chapter 10 Radicals, Radical Functions, and Radical Exponents
81. Perimeter:
22
2 125 2 2 20
2255445
Plw
82. Perimeter:
22
24 20 2 80
845 2165
Plw
83. Perimeter:
45 80 125
Pabc
84. Area:
112
2
1
2
1
()
6(8 2)
Ahbb
(40) (10) 5 40 34.1 5 10 34.1
5 4 10 34.1 5 10 34.1
5210 510
10 10 5 10
ff
22
2
1
f
f
Rcc
v
Rc
Section 10.4 Adding, Subtracting, and Dividing Radical Expressions
87. – 93. Answers will vary.
94.
495xxx
95.
16 9 7xx x
96.
8210xx x
97. makes sense
98. makes sense
99. does not make sense; Explanations will vary.
100. makes sense
101. false; Changes to make the statement true will vary.
A sample change is:
5525
102. false; Changes to make the statement true will vary.
A sample change is:
43 53 93
does not guarantee that the index and radicands
match.
104. false; Changes to make the statement true will vary.
A sample change is: The square root of a negative
number is not a real number,
42.
223252
26252
22
x
x
x
The irrational number is
22.
45
34
24 33 412 5
34 43 112
8948
Chapter 10 Radicals, Radical Functions, and Radical Exponents
107.
22
35 91 3591
649 649
xy ab xyab
108.
23 1 4 2 6
xxx
109.
22
812 6 2xxyyxyxy
110.
22
23
56 69
x
xx xx
111. a. 7( 5) 7 35xx
b.
75735xx
112. a.
2
56 3 6 33 15xx xx
113.
24 24
55
324 324
55 55
24
5
8108
10
2
xy y xy
y
xy
2.
57 46 23
84222
xy xy xy xy xy
3.
33 3 3
22 2 2
34 24 3 2 4 5 4
xx x x
98
1
6.
1/3 1/3 1/3
1/3
3 1/4 3 1/4
3 1/3 1/4 1/3
3
64 64
64
xy x y
xy
Mid-Chapter Check Point
8.
332
444
22
500 500 100 5
444
10 5 5 5
2
xxxx
yyy
xx xx
yy
2
2
12.
1/5
10 10
520 20
1/5
10
1/5
20
10 1/5 2
4
20 1/5
xx
yy
x
y
xx
y
y
15.
36
77
36
7
22
22
xx
xx
16.
11 17 8 16 3
44
816 3
44
24 3
4
32 16 2
16 2
22
x y xy xy
xy xy
xy xy
57
7
722
7
22
7
xy
xy
xxy
xxy
19.
2/3
2/3 2
3
2
11
125
125 125
11
25
5
22.
235 235
47
32 2 32 2
64
xy x y xy x y
xy
Chapter 10 Radicals, Radical Functions, and Radical Exponents
23.
43 6
42 6
46 724
46746
xxy y xy
xxy y y x y
24.
30 5
fx x
To find the domain, we set the radicand greater than or equal to 0.
10.5 Check Points
1. a.
6106 610
660
6415
6215
xx
x
x
x
Section 10.5 Multiplying with More Than One Term and Rationalizing Denominators
2. a.
22 2
56 5 256 6
5230 6
11 2 30
3. a.
3372121
7
77749
b.
33 3333
3
33
33 3
22 3
22 2 23 6 6
93
93
33 3
c.
24 324
55
66
82
xx
xy xy
23
5
324 23
55
2
6
22
xy
x
xy xy
5.
18 18 2 3 3
23 3 23 323 3
22
36 3 54
233
6.
37 37 5 2
52 5252
3 5 3 2 35 14
52
3 5 3 2 35 14
3
xx
10.5 Concept and Vocabulary Check
1. 350;
42 10
;
30 10
;
36
Chapter 10 Radicals, Radical Functions, and Radical Exponents
10.5 Exercise Set
1.
27227
214
xx
x
5.
34 6 2 3
346 323
418 29
8.
3333 3
3674 18712
9.
32
33
32
333
33
22
3
16
16
82
xxx
xxxx
xxx
11.
5262
56 5 2 6 2 2 2
30 5 6 2 2
54 24 5 9 5 4 5
54 24 9 5 20
34 15 5
117 36 7
117 36 7
16.
2
727537
35 21 7 10 7 6 7
35 21 10 7 6 7