Chapter 9
RADICAL EXPRESSIONS,
FUNCTIONS, AND EQUATIONS
9.1 Radical Expressions
and Functions
Exercises
2. The symbol stands for the positive or
principal square root.
4. The square root of a negative number is not
12. 2
49 7 7==
14. 2
144 12 12−=−=−
16. 9−is not a real number.
18.
(
)
2
325 35 35 15===
30.
3
3
327 3 3 3
64 4 4 4
⎛⎞ ⎛⎞
⎟⎟
⎜⎜
−=− =−=−
⎟⎟
⎜⎜
⎟⎟
⎜⎜
⎝⎠ ⎝⎠
32.
()
2
1.21 1.1 1.1==
34. 17 4.123≈
36. 59 7.681≈
38. 0.006 0.077≈
48.
(
)
2
86 43 43
33 3uv uv uv==
50.
(
)
()
2
412 26
26
26
11
64 8
44
18
4
2
ab ab
ab
ab
−=−
=−
=−
Chapter 9 Radical Expressions, Functions, and Equations
158
60.
(
)
(
)
2225
45
9
3
f=+
=+
=
=
)
66.
()
3
3
3
3
28 28 4
24
83
23
g−=−+
=−
=−⋅
=−
68.
()
4gx x=−
x
()
4gx x=−
(
)
,xy
)
70.
()
2gx x=−
x
()
2gx x=−
()
,xy
2
()
22200g=−= =
(
)
2,0
3
(
)
33211g=−==
()
3,1
72. 55
532 2 2−=−=−
74.
3
3
3125 5 5
216 6 6
⎛⎞
⎟
⎜
−=−=−
⎟
⎜⎟
⎜
⎝⎠
76. 336 3.302≈
3
3
3
54
27 2
32
=
=⋅
=
82.
()
9gx x=+
x
()
9gx x=+
()
,xy
5−
()
55942g−=−+= =
()
5, 2−
Section 9.2 Rational Exponents
159
82. (continued)
Mindstretchers
1. We graph points whose coordinates are real
numbers. Since x must be a nonnegative real
number, all values of the square root of x
will also be nonnegative real numbers.
Therefore, all ordered pairs solutions
(
)
,xy
9.2 Rational Exponents
Exercises
2. The expression mn
a is equal to nm
a
and
(
)
.
m
na
5
=−
12. 15 5
yy−=−
14.
(
)
()
12
44
2
2
25 25
5
nn
n
=
=
Chapter 9 Radical Expressions, Functions, and Equations
160
22.
(
)
(
)
()
()
3
35 5
10 10
3
5
2
5
3
2
6
32 32
2
2
8
xx
x
x
x
−=−
⎛⎞
⎟
⎜
=− ⎟
⎜⎟
⎜⎟
⎝⎠
=−
=−
26.
13
13 9
9
9
3
3
3
3
3
8
8
8
2
2
y
y
y
y
y
−⎛⎞
⎛⎞ ⎟
⎟⎜
⎜⎟
⎟=⎜
⎜⎟
⎟⎜
⎜⎟
⎟
⎜⎟
⎜
⎝⎠ ⎝⎠
=
⎛⎞
⎟
⎜⎟
=⎜⎟
⎜⎟
⎟
⎜
⎝⎠
=
16
1
4
=
=
34.
(
)
()()
32 16 32
16
14
4
81 81
81
81
=
=
=
4
x
=
40.
()
()
1
66
3
3
3
2
3
2
27 27
3
3
yy
y
y
=
=
=
42.
56 56 23
23
16
7
7
pp
p
−
=
Section 9.2 Rational Exponents
161
46.
(
)
(
)
2
23 3
29 2 9
264 2 64
nn n n
−=−
48.
()
()
33
434 4
4
34
3
3
12 12
3
3
12
81
4
27
xx
x
x
x
x
⋅
=
=
=
50.
(
)
12
66
81 81
xx
=
54.
13
22
3
33
32
3
3
xx
yy
x
y
⎛⎞
⎟
⎜⎟=
⎜⎟
⎜⎟
⎟
⎜
⎝⎠
=
58. 16 16
35.74 0.62 35.75 0.43WTSTS=+− +
b.
()
(
)
3
3
0.4 0.4 36
0.4 46,656 0.4 216 86.4
D=
===
It takes Mercury about 86 Earth days
to revolve once around the sun.
Mindstretchers
1.
(
)
3
1.5 3 2 3
99 9327== ==
c.
()
()
(
)
(
)
113
13
22
3
3
xy xy
xy
xy xy
−
+=+
+
=+ = +
d.
(
)
()
13
223
10 25 5
aa a
−+ −
=−
Chapter 9 Radical Expressions, Functions, and Equations
162
9.3 Simplifying Radical
Expressions
Exercises
2. In general, the expression na is not
simplified if the radicand a has a factor
that is a perfect nth power.
4. The distance between two points
(
)
11
,xy
and
()
22
,xy on a coordinate plane is equal
10. 8
14 18 14 18 38 3
84 ttt t t t t
+
⋅= ⋅= = =
12. 5
ab⋅ Cannot be simplified.
14.
12
12 16 13 3
16
6
nn nnn
n
n
−
== ==
16.
(
)
13
12 1213 1636
xx x x x
⋅
====
18. 63
4 12 46 126 23 2 2 2
uv u v u v v u===
20. 84
4 2 48 24 12 12
nnnn n n
+
⋅=⋅= =
32. 22 22 4
55 5 5
64 64 24yy yy y⋅=⋅=
34. 10 10 2
55
qqq
ppp
⋅= ⋅=
36. 63 9 7 9 7 3 7=⋅=⋅=
38. 598 5492 549 2
57 2 35 2
=⋅=⋅
=⋅ =
48. 72 36 2 36 2 6 2aa aa=⋅=⋅ =
50. 66 6 3
216 36 6 36 6 6 6tt t t=⋅=⋅=
52. 79 68
68 34
125 25 5
25 5 5 5
ab ab ab
a b ab a b ab
=⋅
=⋅=
54. 33 33
86262
3
22
108 27 4 27 4
34
xxxxx
xx
=⋅=⋅
=
56. 98 8
44 4
4
243 81 3 81 3
yyyyy
=⋅=⋅
Section 9.3 Simplifying Radical Expressions
163
64. 65 65 5
13
13
yyy==
72. 49 49 7
64 8
64
==
80. 63
6
55
5
36 6
36
pp
p
qq
q
==
82. 42
4
442
81 9
81
xxx
yy
y
==
92.
()()
22
21 21
22
dxx yy=−+−
()()
()()
22
22
19 82
10 10
=−− +−−
=− +−
()()
22
43
16 9
25
5 units
=− +−
=+
=
=
98.
322
3
3
63 63 9
xx
x
Chapter 9 Radical Expressions, Functions, and Equations
164
106.
()()
22
21 21
dxx yy=−+−
108. 2
32 16 4
16
hh hh
===
110. 0.75 8
px
=−
112. 22
450 400
202,500 160,000
d=+
=+
114. a. 4 blocks west would correspond to
4,x=− and 6 blocks south would
correspond to 6.y=− The friend’s
apartment is located at
(
)
4, 6 .−−
b.
()( )
22
21 21
dxx yy=−+−
Mindstretchers
()()
()
()
()()()
2
2
11 1
11
111
xx x
xx
xxx
=+−+
=+ −
=++−
56
xx
++
()()
()()
229 2
23
xx x
xx
+− +
=++
9.4 Addition and Subtraction
of Radical Expressions
Exercises
2. Like radicals can be added by combining
their coefficients and then multiplying this
sum by the radical.
4. The Pythagorean theorem can be expressed
as 22
.cab=+
6.
(
)
27 57 2 5 7 77+=+ =
)
)
16.
(
)
813 3 8133
2
aaa a
a
− + =−+
=−
18.
(
)
33 3 3
3
11 15 11 1 15
3
xx x x
x
−−+ =−−+
=
28.
()
12 75 4 3 25 3
23 53
25 3
33
−=⋅−⋅
=−
=−
=−
30.
()
628 263 77
647 297 77
62 7 23 7 7 7
12 7 6 7 7 7
12 6 7 7
13 7
−+−
=− ⋅ + ⋅ −
=− ⋅ + ⋅ −
=− + −
=− + −
=−
34.
()
3
2
32 98
16 2 49 2
42 72
47 2
32
yy y
yy yy
yy yy
yy y
yy
−
=⋅− ⋅
=−
=−
=−
Chapter 9 Radical Expressions, Functions, and Equations
166
42.
(
)
27 22
55
27 22
55
522 22
55
22 22
55
22
5
10 2 4 64
10 2 4 32 2
10 2 4 2 2
10 2 8 2
10 8 2
xy xy
xy xy
yxy xy
yxy xy
yxy
+
=+⋅
=⋅+⋅
=+
=+
46.
()
()
()
333
33
33
33
3
3
32
192 6 24 216
23
32
64 3 6 8 3 6
23
3436234
2
63 123 4
612 34
63 4
−++−
=− ⋅ + ⋅ + −
=− + ⋅ −
=− + −
=− + −
=−
()
33
22 22
3
222
3
22
36 156
315 6
12 6
xx xx
xxx
xx
=−
=−
=−
()
(
)
33
822
33
62 2 2
() 36 5 162
365276
fx gx x x x
xx x x
−= −−
=⋅+ ⋅
(
)
33
3
22 32
232
bab ab
bab
=−
=−
54. 3
65rr−+ Cannot be combined.
56.
()
24
24
2
22
22
() 2 63 97
29 79 7
23 79 7
6797
69 7
fx gx x x x
xx x
xx x
xx
xx
+=− −
=− ⋅ − ⋅
=− ⋅ −
=− −
=− −
Section 9.4 Addition and Subtraction of Radical Expressions
167
60. side = 52,900 230=
()
Perimeter = 4 side
4230
920
×
=
=
The perimeter of the base is 920 m.
)
)
)
)
2.
(
)
()
(
)
918
48
52
fx x
gx x
hx x
=+
=+
=+
a.
(
)
(
)
292181818366
f
=+=+==
(
)
(
)
(
)
(
)
3. Yes, if a or b is 0. If 0a= and 0,b≥
then 00bbb+=+= and
0.bb+= If 0a≥ and 0,b= then
00aaa+=+= and 0.aa+=
Chapter 9 Radical Expressions, Functions, and Equations
168
9.5 Multiplication and Division
of Radical Expressions
Exercises
2. When squaring binomials that contain
radical terms, apply the formulas for
squaring a binomial.
8. 63 63 18 9232⋅=⋅= =⋅=
10.
(
)
(
)
11 62 6112 622−=−⋅=−
14.
(
)
(
)
53 53
8
8
4
4
412 36 4312 6
12 72
12 36 2
12 6 2
72 2
yy yy
y
y
y
y
=⋅ ⋅
=
=⋅
=⋅
=
18. 32 26 32 26
44 4
58
4
48
4
24
850 850
400
16 25
225
xy xy xy xy
xy
xy x
xy x
−⋅ =−⋅
=−
=− ⋅
=−
20.
(
)
5203 520 53
100 3 5
+= ⋅ + ⋅
=+
36
=−
24.
(
)
()
()
2
3227 42
32 2 7 3 4 2
−−
=− ⋅ ⋅ − −
44
44
44
64 7 40
16 4 7 40
24 740
=− +
=− ⋅ +
=− +
28.
(
)
22
33
3
3
yyy yyyy
yy yy
yy
−=⋅−⋅
=⋅−⋅
=−
Section 9.5 Multiplication and Division of Radical Expressions
169
32.
(
)
(
)
)
)
)
51 56
F 0 I L
+−
34.
(
)
(
)
)
)
)
25 125 1
F O I L
−+ −
36.
(
)
(
)
2
5101020
F O I L
5 10 5 20 10 10 20
20 15 2
++
=⋅ +⋅ + + ⋅
=+
38.
(
)
(
)
()
2
2735
F O I L
6102135
611 35
pp
ppp
pp
+−
=−+−
=+ −
46.
(
)
(
)
(
)
22
34 34 3 4
316
yy y
y
+− ++ = + −
=+−
50.
(
)
(
)
(
)
()
22 2
82 8 2822
84424
aaa
aa
+= + +
=+ ⋅+
54.
108 1 108
23
23
136
−=−
=−
56.
3
3
48 48
3
3
16
4
aa
a
a
a
a
=
=
=
58.
55
108 108
36
36
yy yy
y
y
−=−
Chapter 9 Radical Expressions, Functions, and Equations
170
60.
388
34
34
256 1 256
82
82
xx
xx
xx
=
62. 222
49 7
49
xxx
==
64.
222
422
4
18 18 9 2 3 2uuuu
vvv
v
⋅
== =
70. 2
3 3 63636 6
62
6666
=⋅= = =
72. 77 7
37 21
63 3 7 7
uu uu
=⋅= =
⋅
80. 77 7
2
42
pq pq pqr
r
r
rrr
=⋅=
84. 33 3nn
nn
nn
=⋅=
()
2
2
2
2
2
32
72 2
6
72
6
uv
vv v
uv
vv
uv
=⋅
=
=
()
2
3
33
2
3
2
3
9
327
9
33
9
pqr
r
pqr
r
pqr
=
=
=
94.
(
)
2
2
xyx
xy
x
xx x
+
+⋅=
Section 9.5 Multiplication and Division of Radical Expressions
171
96.
()
()
2
2
43 2 6 43 6 26
66 6
418 12
6
43 2 2 3
6
26 2 3
6
62 3
3
nnnnn
nn n
nn
n
nn
n
nn
n
nn
n
−−
⋅=
−
=
⋅−
=
−
=
−
=
100.
(
)
()
22
351
351
51 51 51
35 3
51
35 3
4
−
−
⋅=
+− −
−
=−
−
=
104.
(
)
()
()
()
()
()
2
2
96 3
63
9
6363 63
96 3
36 3
96 3
312
36 3
12
18 3 3
y
y
yy y
y
y
y
y
y
y
y
+
+
⋅=
−+ −
+
=−
+
=−
+
=−
+
2
ab b
ab
+
=−
108.
()
()()
2
22
15
55
55
5
nn
nn
nnn
n
−+
⋅
−+
+−−
=
−