y
⫺4⫺224
⫺5⫺3⫺1135
⫺1
x
y
y
Chapter 10
Radical Expressions, Equations, and
Functions
Exercise Set 10.1
The domain is [−9,∞). The answer is (b).
3≥x
The domain is (−∞,3]. The answer is (g).
6. 9,−9
18. 6.528
24. t(x)=−√2x+1
t(−1) does not exist as a real number.
26. F(x)=√x2+1
F(0) = √02+1=√1=1
28. Solve: √2x+1≥0
30. B=h×w
3600 =165 ×63
3600 ≈1.7m
2
32.
34.
36.
y
x
y
⫺
4
⫺
224
⫺
5
⫺
3
⫺
1135
⫺
5
⫺
1
⫺4
⫺3
y
268 Chapter 10: Radical Expressions, Equations, and Functions
38.
40.
42.
44. √25t2=(5t)2=|5t|=5|t|
52. −4
54. −5y
√−1000 = −(−10)=10
60. g(x)=−3
√2x−1
62. g(t)= 3
√t−3
64. −4
√256 = −4
76. 1999
(2a+b)1999 =2a+b
82. x2+x=0
84. 2x2−26x+72=0
2(x−4)(x−9)=0
88. x3−x2=0
x=0or x =1
Exercise Set 10.2
RC2. (b)
RC6. (d)
√4)7=2
7= 128
Exercise Set 10.3 269
16. 7
x3y2z2=(x3y2z2)1/7
24. 16−3/4=1
163/4=1
(4
√16)3=1
23=1
8
38. 6a
4
√b=6a
b1/4
44. 35/8
3−1/8=3
6/8=3
3/4
54. (a−3/2)2/9=a−1/3=1
a1/3
60. 6
√t4=t4/6=t2/3=3
√t2
62. 4
√a12 =a12/4=a3
√a−10 =a−10/5=a−2=1
72. 4
16x4y2=4
24x4y2=(2
4x4y2)1/4=2xy1/2=
(116·137)1/42 =42
√116·137
78. 3
√y5
√3y=y1/3(3y)1/5=y5/15(3y)3/15 =
84. a2/3·b3/4=a8/12 ·b9/12 =(a8b9)1/12 =12
√a8b9
90. |3x|= 120
The solution set is {−40,40}.
1
94. 10
5
√x1555
10
√x155
=x3
Exercise Set 10.3
270 Chapter 10: Radical Expressions, Equations, and Functions
4. √18 = √9·2=3
√2
14. 4
√32 = 4
√16 ·2=24
√2
16. 75p3q4=25p2q4·3p=5pq2√3p
24. √2√32 = √2·32 = √64 = 8
30. √5a7√15a3=√75a10 =√25a10 ·3=5a5√3
40. 30x3y418x2y5=30x3y4·18x2y5=
42. √63
√5=6
1/2·51/3=6
3/6·52/6=(6
3·52)1/6=
46. 3
√x26
√x5=x2/3·x5/6=x3/2=x1+1/2=x·x1/2=x√x
52. 4
√9ab3√3a4b=(9ab3)1/4(3a4b)1/2
54. √98
√2=98
60. √52ab3
√13a=52ab3
13a=√4b3=√4b2·b=2b√b
2√3=1
√64a11b28
64a11b28
√x
√a7b−4,or 15
a7
86. 3
64x7
3
√64x6·x
216y6=4x23
√x
√x
Exercise Set 10.4 271
92. 5
32b10
5
√32b10
√243a20 =2b2
98. Let h= the height. Then h+ 2 = the length of the base.
100. 4x
x+5+20
x=100
x2+5x
x+5+x(x+5) ·20
x= 100
x·4x+ 20(x+ 5) = 100
4x=0 or x +5=0
102.
3
x3−y3
3
√x−y=3
Exercise Set 10.4
10. 6√7+ 4
√11
18. 3
√27 −53
√8=3−5·2=3−10 = −7
24. 3
√54x−3
√2x4=3
√27 ·2x−3
√x3·2x=
32. 4
√80x5−4
√405x9+4
√5x
34. √9y+27+√y+3=9(y+3)+√y+3=
38. 2√6+6
46. 3−43
√63
272 Chapter 10: Radical Expressions, Equations, and Functions
60. (√6−√7)(√6+√7) = 6 −7=−1
72. (2 −√x)(1 −√x)=2−2√x−√x+x=2−3√x+x
√7+ 3
√6)(2 3
√7−33
√6) =
78. (3
√8x−3
√5y)2=(23
√x−3
√5y)2=
43
√x2−43
√5xy +3
25y2
80. a2−4
a÷a−2
a+4 =a2−4
a·a+4
a−2
82. y3−27
y2−9·y2−6y+9
y2+3y+9
=(y−3)(y−3)
y+3
84. 1−1
x
=
1−1
x
·x2
86.
1
a+1
b
=
1
a+1
b
·a3b3
88. |3x+7|<22
90. |3x+7|=|2x−5|
3x+7=2x−5or 3x+7= −(2x−5)
5
92. Left to the student
94. (√x+2−√x−2)2=(√x+2)2−2√x+2√x−2+(√x−2)2=
x+2−2√x2−4+x−2=2x−2√x2−4
98. 3+2+√14
=3+√2+1
Chapter 10 Mid-Chapter Review
1. False; negative numbers do not have real-number square
6. 5√32 −3√18 = 5√16 ·2−3√9·2
f(x) 2
冑
x
y
Chapter 10 Mid-Chapter Review 273
10. √−9 does not exist as a real number because negative
11. f(x)=√2x+3
12. The domain of f(x)=√4−xis the set of all x-values for
13. Graph f(x)=−2√x.
1−2 (1,−2)
14. Graph g(x)=√x+1.
x g(x) (x, g(x))
15. √36z2=(6z)2=|6z|=6|z|
√27a3=−3aSince (3a)3=27a3, then 3
√27a3=3a
20. 10
y10 =|y|
24. 3
√6m2n=(6m2n)1/3
25. 31/4·3−5/8=3
33/8
28. (n−3/5)5/4=n−3/5·5/4=n−3/4=1
32. a2/3b3/5=a10/15b9/15 =(a10b9)1/15 =15
√a10b9
3
√80
80
√40 = 3
√8·5= 3
√83
√5=23
√5
36. 49a5
b8=√49a5
√b8=√49 ·a4·a
√b8=√49a4√a
√b8=7a2√a
b4
37. 5√7+6
√7=(5+6)
√7=11
√7
39. √3(2 −5√3) = 2√3−5√9=2
√3−5·3=2
√3−15
44. Yes; since x2is nonnegative for any value of x, the nth
45. Formulate an expression containing a radical term with an
46. Since x6≥0 and x2≥0 for any value of x,3
√x6=x2.
274 Chapter 10: Radical Expressions, Equations, and Functions
Exercise Set 10.5
2. 8
7=8
7·7
7=√56
7=√4·14
7=2√14
7
8. 3
3
3
3
√9
14. 1
3
√yz =1
3
√yz ·
3
y2z2
3
y2z2=
3
y2z2
7c
7c
3
√70a2bc
20. 2x
5
18x8y6=2x
5
2·32·x8y6=
5
24·33·x2y4
22. 3
8+√5=3
8+√5·8−√5
8−√5=24 −3√5
64 −5=
24. −5√2
√7−√5=−5√2
√7−√5·√7+√5
√7+√5=−5√14 −5√10
7−5=
26. 34√5
2√5−√3=34√5
2√5−√3·2√5+√3
2√5+√3=
28. 2+ √3
√3+√5=2+ √3
√3+ √5·√3−√5
√3−√5=2√3−2√5+ 3−√15
3−5=
32. √6−3√5
√3−2√7=√6−3√5
√3−2√7·√3+2
√7
√3+2
√7=
34. 5+√x
8−√x=5+√x
8−√x·8+√x
8+√x=
38. 7√2+4
√3
√3
√2
√2=
√ab +b
42. 5
x−1+9
x2+x+1 =15
x3−1,
LCM is (x−1)(x2+x+1)
x=−19
5or x=1
Only −19
=(2x+3)✏(x−2)
46. a) x−5=(
√x+√5)(√x−√5)
√x+√a)(√x−√a)
Exercise Set 10.6 275
48. 1
4+√3+1
√3+1
√3−4=
−13√3·√3
√3=7√3
39
Exercise Set 10.6
RC4. A radical equation has a variable in one or more
radicands.
This value checks.
4. √3x−4=6
This value checks.
6. √x−1−3=9
8. √2y+9=5
This value checks.
10. 3
√y=−2
y=−8
12. √y−3=−2
14. 3
√x−2=3
x−2=27
18. 3
√3y+6+2= 3
3y+6=1
3y=−5
20. 3= 1
√y
22. x−5=√x+7
x2−10x+25 =x+7
24. √2x+7−2=x
√2x+7= x+2
26. x−1=√x+5
x=4 or x =−1
Only 4 checks. It is the solution.
28. x−1=√1−x
x2−x=0
276 Chapter 10: Radical Expressions, Equations, and Functions
30. √5x−3=√2x+3
32. √x−9+√x=1
5=√x
34. √4x−3=2+√2x−5
√2x−5+2x−5
x2−10x+21=0
36. 4+√10 −x=6+√4−x
1=16−4x
This value checks.
38. √y+15−√2y+7=1
−y+7=2
√2y+7
40. √6x+7−√3x+3=1
9x2+6x−3=0
42. √2m−3=√m+7−2
m2−28m+ 196 = 16m+ 112
44. 3√2y+3−√y+10=0
17y=−17
48. 13=1.2√h
50. 230 = 1.2√h
230
52. At 65 mph: 65 = 2√5L
32.5=√5L
Exercise Set 10.6 277
54. 1113 = 21.9√5t+ 2457
The temperature was approximately 25◦F.
56. 2 = 2(3.14)L
32
58. 3.1 = 2(3.14)L
32
9.61 = 39.4384 ·L
32
60. Let t= the time it takes Jeff to do the job. Then 3t=
62. Let h= the number of hours Dharma would have to work
in order to earn $1044.96.
64. 3x2−5x=0
2x2−x−21=0
68. f(a+h)−f(a)
70. f(a+h)−f(a)
72. We can graph y1=√2x+1+
√5x−4 and y2=√10x+9.
The first coordinate(s) of the point(s) of intersection are
the solution(s) of the equation. There is one point of in-
tersection, (4,7), so the solution is 4.
x=−4or x =3
The numbers −4 and 3 check and are the solutions.
76. 6√y+6y−1/2=37
6y+6=37
√y
36 or y =36
The numbers 1
5√x=x
80. √x+1−2
√x+1 =1
x2−3x=0
278 Chapter 10: Radical Expressions, Equations, and Functions
82. 2√x−1−√3x−5=√x−9
2√3x2−8x+5=3x
84. Let y=7+4
√3−7−4√3.
Exercise Set 10.7
2. c2=8
2+10
2
4. c2=8
2+8
2
6. 52+b2=12
2
8. (4√5)2+b2=10
2
10. 12+b2=(
√12)2
12. (√n)2+b2=2
2
16. Solve d2=65
2+65
2.
w=√6,980,904 ft
ADC
30 ft
h=√400
PSR
40 ft
h2= 225
Exercise Set 10.8 279
22.
✻
y
(0,y)❅❅❅❅❅
q
(3,0)
The points are (0,−4) and (0,4).
24.
d2= 4900 + 8100
28. Let s= the length of a side of the square.
30. h2=s2+s2
h2=2s2
h=s√2
32. Let r= the speed of the current. Then 3r= the speed of
34. x2+24=11x
36. 3x2−12=0
3(x2−4)=0
38. x−1
x−3=6
x−3,LCM is x−3
40. a) Tw=35.74+0.6215(40) −35.75(25)0.16+
0.4275(40)(25)0.16
≈29◦F
c) Tw=35.74+0.6215(10) −35.75(20)0.16+
d) Tw=35.74+0.6215(10) −35.75(40)0.16+
f) Tw=35.74+0.6215(−15) −35.75(35)0.16+
Exercise Set 10.8
RC2. True; see page 753 in the text.
6. −√−20 = −√−1·4·5=−2i√5, or −2√5i
280 Chapter 10: Radical Expressions, Equations, and Functions
18. −√−76 + √−125 = −√−1·4·19 + √−1·25 ·5=
32. √−16 ·√−64 = 4i·8i=32i2=−32
50. (6+2i)2=36+24i+4i2=32+24i
60. (−i)71 =(−1)71(i)71 =−1·i70 ·i=−(i2)35 ·i=−(−1)35 ·
68. i29 +33i=i28 ·i+33i=(i2)14 ·i+33i=
(−1)14 ·i+33i=i+33i=34i
70. 5i5+4i3=5·i4·i+4·i2·i=5·(i2)2·i+4·(−1) ·i=
88. 8i
96. x2−2x+5= 0
(1+2i)2−2(1+2i)+5 ? 0
98. x2+2x+2 =0
100. x-intercept: 4 ·0−3x=72
−3x=72