Chapter 10
Radicals, Radical Functions, and Rational Exponents
10.1 Check Points
1. a. 64 8because
2
864
b. 49 7because
2
(7) 49
2. a.
() 12 20
(3) 12(3) 20
36 20
16
4
fx x
f
3.
() 9 27
fx x
9270
927
3
x
x
x
Domain of f is
3xx or [3, ).
5. a.
2
(7) 49 7
b.
2
(8) 8xx
10 5
3
b.
3
3
3
() 2 2
(5) 2(5) 2
8
2
gx x
g
d.
5
11
9. a.
4
4
(6) 6xx
b.
5
5
(3 2) 3 2xx
Chapter 10 Radicals, Radical Functions, and Radical Exponents
4.
5200x
5.
a
6.
3
10
7.
3
(5)
10.1 Exercise Set
1.
36 6
because
2
636
2.
16 4
because
2
416
7.
11
25 5
because
2
11
525
10.
42
25 5
because
2
24
525
11.
0.81 0.9
because
2
0.9 0.81
12.
0.49 0.7
because
2
0.7 0.49
not a real number
20.
25 144 119
not a real number
21.
2fx x
18 18 2 16 4f
1134f
not a real number
Section 10.1 Radical Expressions and Functions
23.
23gx x
11 2 11 3
22 3
g
24.
42418193g
12112131.73g
25.
2
1hx x
22
551 444h
26.
22
552 333h
22
332 111h
27. To find the domain, set the radicand greater than or
equal to zero and solve.
30
3
x
x
The domain of f is
3, .
This corresponds to
The domain of f is
5, .
This corresponds to
graph (d).
30.
3150
x
62 0
26
3
x
x
x
|
This corresponds to graph (e).
32. 82 0
28
x
x
Chapter 10 Radicals, Radical Functions, and Radical Exponents
35.
2
444
36.
2
10 10 10
40.
2
4222
81 9 9 9xxxx
41.
2
63
33
100 10
10 10
xx
xx
46.
2
2
10 25 5 5xx x x
47.
3
27 3 because
3
327
48.
3
3
64 4 because 4 64
51.
3
11
125 5
because
232232
77 7
711 711
xy xy xy xy
53.
3
because
1000 10 10 1000
54.
3
3
82 2 8
because
125 5 5 125
55.
3
1fx x
33
28 28 1 27 3f
57.
3
88gx x
3
3
3
2828168
82
g
3
3
3
181888
000
g
Section 10.1 Radical Expressions and Functions
60.
5
5
1 1 because 1 1
61.
4
4
16 2 because 2 16
66.
4
81
not a real number
67.
5
5
1 1 because 1 1
68.
7
7
1 1 because 1 1
69.
6
1
not a real number
75.
5
5
32 2 because 2 32
76.
6
6
64 2 because 2 64
77.
33
xx
81.
33
82xx
82.
33
125 5xx
86.
6
6
666
87.
4
4
33xx
88.
4
4
55xx
89.
5
5
32 1 2 1xx
5
Chapter 10 Radicals, Radical Functions, and Radical Exponents
92.
2
0002022
xfxx
f
Range:
2,
93.
3
333300
444311
xfxx
f
f
94.
4
554593
xfxx
f
95. The domain of the cube root function is all real
numbers, so we only need to worry about the square
root in the denominator. We need the radicand of
the square root to be
0
, but we also cannot divide
by 0. Therefore, we have
The domain of f is
,16
.
97. From the numerator, we need
10x
. From the
denominator, we need
30x
. We need to solve
the two inequalities. The domain of the function is
the overlap of the two solution sets.
Section 10.1 Radical Expressions and Functions
98.
20x
. From the denominator, we
need
70x
. We need to solve the two
99.
333
4
16 625 2 25 27 3
101. a.
( ) 2.9 20.1
(48) 2.9 48 20.1 40.2
fx x
f
The model estimates the median height of boys
who are 48 months to be 40.2 inches. This
underestimates the actual median height by 0.6
inches.
c. Find (50)f and (60).f
( ) 2.9 20.1
(50) 2.9 50 20.1 40.6
fx x
f
102. a.
() 3.1 19
(48) 3.1 48 19 40.5
fx x
f
(10) 3.1 10 19 28.8
f
Average rate of change is
(50) 3.1 50 19 40.9
(60) 3.1 60 19 43.0
f
f
Average rate of change is
(60) (50) 43.0 40.9 0.2
60 50 10
ff
m
inches
per month.
This is a much smaller rate of change.
This is shown on the graph because the graph is
104.
45 20 45 900 30f
The officer should believe the motorist. The model
predicts that the motorist’s speed was 30 miles per
Chapter 10 Radicals, Radical Functions, and Radical Exponents
115.
12
3
4
3
yx y x
yx
116.
yx
4yx
3yx
117.
fx x gx x
hx x kx x
118.
2
1
2
yx
yx
b.
2
xx
for (0, ).
119. does not make sense; Explanations will vary.
Sample explanation: Because the negative is raised
to an even power first, this expression will simplify
to positive 8.
120. does not make sense; Explanations will vary.
cube root, the radicand is not required to be greater
than zero.
124. false; Changes to make the statement true will vary.
A sample change is: If b is negative and n is even,
the expression does not represent a real number.
However, this does not hold when n is odd.
Section 10.1 Radical Expressions and Functions
126. false; Changes to make the statement true will vary.
A sample change is: As the index n gets larger, the
value of the radical will get smaller.
the radicands are greater than zero and the
denominator does not equal zero.
30and 10
31
xx
xx
The domain of
f
g
is
3, .
131.
3
hx x
the left.
132.
32 3 5 32 315
32215
3430
730
xxx xxx
xx
xx
x
134.
3411
3 4 11 or 3 4 11
37 315
75
3
x
xx
xx
x
x
Chapter 10 Radicals, Radical Functions, and Radical Exponents
10.2 Check Points
1. a.
1/2
25 25 5
2. a.
1/4
455xy xy
b.
1/5
33
5
22
ab ab
5. a.
1/2
1/2
111
100 10
100 100
b.
1/3
1/3 3
111
82
88
c.
3/5
3/5 3 3
5
1111
32 8
32 2
32
d.
11
331111
11
312
553435
412
1
y
xy x y xy
b.
312 12 4
33
3
88 8 2
aa a a
c.
1
42 42
88
11
42
88
xy xy
e.
1111
2
36
3326
xx x x x
10.2 Concept and Vocabulary Check
1. 36; 6
2. 8; 2
Section 10.2 Rational Exponents
10.2 Exercise Set
1.
1/2
49 49 7
2.
1/2
100 100 10
3.
1/3 3
27 27 3
7.
1/3 3
xy xy
8.
1/4 4
xy xy
13.
2
2/3 2
3
125 125 5 25
14.
2
2/3 2
3
1000 1000 10 100
15.
3
3/5 3
5
32 32 2 8
19.
4
4/7 4
7
7
or
xy xy xy
20.
4
4/9 4
9
9
or
xy xy xy
21.
1/2
77
25.
11 11
xx
26.
1/5
5
13 13
xx
17 15 2 15 2
31.
1/5
22
5
xy xy
32.
1
33
77
xy xy
33.
33/2
19 19
xy xy
Chapter 10 Radicals, Radical Functions, and Radical Exponents
39.
1/2
1/2
111
49 7
49 49
40.
1/2
1/2
111
93
99
44.
5/4
5/4 5 5
4
1111
81 243
81 3
81
45.
2/3
2/3 2 2
3
1111
84
82
8
49.
2/3
2/3 2
3
2
11
64 64 64
11
16
4
51.
7/10
7/10
7
7
10 10
1
22
11
or
2
2
xy
xy
xy
xy
4/4 1
333
56.
2/3 1/3
2/3 1/3 3/3 1
55 5 5 55
57.
3/4 3/4 1/4 2/4
1/4
1/2
16 16 16
16
16 16 4
61.
4/5 4/5 1/5 3/5
1/5
xxx
x
62.
3/7 3/7 1/7 2/7
1/7
xxx
x
Section 10.2 Rational Exponents
64.
1/4 1/4 3/5 5/20 12/20
3/5
xxx
x
1/6
yy
68.
1/6
3/4 3/24 1/8
1/8
1
yyy
y
71.
1/2 1/2 1/2
46 1/2 4 6
4 1/2 6 1/2
23
25 25
25
5
xy x y
xy
xy
72.
1/3 1/3 1/3
96 1/3 9 6
32 32
3
125 125
125 5
xy x y
xy xy
75.
1/2 3/4 1/2 3/4 1/4
1/4
33 3
3
9/12 1/12
8/12 2/3
27
27
27 27
y
y
yy
16 16
yy
79.
822/81/4
4
xx x x
80.
10 22/101/5
5
xx x x
81.
361/36/3 2
88 2
aaa
82.
312 1/3 12/3 4
27 27 3
aaa
Chapter 10 Radicals, Radical Functions, and Radical Exponents
91.
63 6/93/9
9
2/3 1/3 2
3
xy x y
xy xy
92.
26 2/46/4 1/23/2 3
4
xy x y x y xy
95.
1/5
5221/2
2/5 1/2 (2/5) (1/2)
4/10 5/10 9/10
10 9
xxx x
xx x
xx
x
96.
7
21 4
21
7272 1414
72
11 14 11
14
xxxx x x
xx
(1/6) (4/6) 2/3 5/6 2/3
6
5/6 4/6 5 4
abab
ab ab
99.
1/4
41/4 1/5
1/5
5
5/20 4/20
xx x
xx
x
101.
1/6
6
4/6 1/6 3/6
1/2
yy
yy
yy
102.
22/5
52/5 3/10 4/10 3/10
3/10
3
10
1/10 10
yyyy
y
y
yy
Section 10.2 Rational Exponents
106.
1/2
1/2 1/2
22 2
1/2 1/2 1/4
222
4
xy xy xy
xy xy xy
109.
12
12 1/4
35 35
4
1/4 12
35
12/4 3
35 35
33 53 9 15
xy xy
xy
xy xy
xy xy
112.
433 3/43/4
1/2 1/2
3/4 1/2 3/4 1/2
3/4 2/4 3/4 2/4
1/4
1/4 1/4
4
ab a b
ab a b
ab
ab
ab ab
ab
113.
1/3 1/3 2/3 1/3 1/3 1/3 2/3
1/3 1/3 1/3 2/3
2/3 3/3
xx x xx xx
xx
xx
115.
1/2 1/2
1/2 1/2 1/2 1/2
1/2 1/2 1/2 1/2
2/2 1/2
1/2
35
53 35
5315
215
215
xx
xx x x
xxx
xx
xx
116.
1/3 1/3
26
xx
1/4
42
xx
119.
1/3 1/3 3/3
1/3 2/3
1/3
1/3 1/3 2/3
1/3 1/3 2/3
1/3 2/3
15 60 15 60
15 60
15 60
15 1 15 4
15 1 4
xxxx
xx
xxx
xxx
xx
Chapter 10 Radicals, Radical Functions, and Radical Exponents
121.
1/2
24 1/2
1/2 1/2
1/2 24 1/2
2 1/2 4 1/2 1/2
1/2
21/2
12 1/2 11
2
23/2
3/2
49
49
1
49
11
77
1
77
xy xy
xyxy
xy xy
xy xy x y
x
xy y
123.
6
5/4 1/3 6
5/4 3/4 1/3
3/4
62/4 6 1/3 6
2/4 1/3
3
32
2
xy xy
x
xy x y
x
xy y
Section 10.2 Rational Exponents
128.
3/4
3/4
70
80 70 70 1694
fx x
f
A person who weighs 70 kilograms needs about 1694 calories per day to maintain life.
131.
4/25 4/25
35.74 0.6215 35.74 0.4275
Ctvtv
a. For
0
t
, we get
4/25
35.74 35.74
Cv v
132.
4/25 4/25
35.74 0.6215 35.74 0.4275
Ctvtv
a. For
30
t
, we get
Chapter 10 Radicals, Radical Functions, and Radical Exponents
133.
3
1.25 9.8 16.296
LSD
a.
1/2 1/3
1.25 9.8 16.296
LS D
134.
3
1.25 9.8 16.296LSD
a.
1/2 1/3
1.25 9.8 16.296LS D
b.
1/2 1/3
1.25 9.8 16.296
LS D
135. – 144. Answers will vary.
145. The simplification is correct.
146. The simplification is not correct.
147. The simplification is not correct.
1/4 1/4 1/2 3/4
1/2 3/ 4
xx
xx
Sample explanation:
77 7 7 7.
149. does not make sense; Explanations will vary.
Sample explanation: It is often easier to find the nth
root before raising the expression to the mth power.
occur before the exponent is evaluated.
152. true
153. false; Changes to make the statement true will vary.
A sample change is:
1/ 1/ 1/
() .
nnn
ab a b
Do not
Section 10.2 Rational Exponents
157.
4
3
4/3 2 4/3 2
3/4 1
3/4 1 3
4
11
11 4
8
82 82
11 1 1
16 2
2
16 2 16
158.
3/2
2/3 2/5 1/2
327 32 9
159. First simplify.
1/2 1/2
1/2 1/2
34
34
fx x x
xx
160.
21
21
31 2 2
45 1
yy
mxx
32 4
yx
Solve for y to write the equation in slope–intercept
form.
c.
16 4 16 4
164. a.
300 17.32
b.
10 3 17.32
Chapter 10 Radicals, Radical Functions, and Radical Exponents
10.3 Check Points
1. a.
5 11 5 11 55
2. a.
80 16 5 16 5 4 5
3.
2
2
2
2
() 3 12 12
3( 4 4)
3( 2)
3( 2)
32
fx x x
xx
x
x
x
4.
9113 8102 45
xy z xy z xyz xyz xyz
10.3 Concept and Vocabulary Check
1.
n
ab
2. 77