chapter 9 radical expressions and negative so this is not defined using 4. 25 121 5 …€¦ ·...
TRANSCRIPT
Chapter 9 Radical Expressions and Equations
Section 9.1. Introduction to Radical Expressions
Practice 9.1.11. Since 42 = 16 16 = 4
2. Since 02 = 0 0 = 0
3. Since (3/8)2 = (9/64) 9 64
= 3 8
4. Since (13)2 = 132 132 = 13
5. Since (0.5)2 = 0.25 0 . 25 = 0 . 5
6. Since (0.07)2 = 0.0049 0 . 0049 = 0 . 07
Practice 9.1.21. 0 . 03 = 0 . 1732 ≈ 0 . 2
2. 2550 = 50. 4975 ≈ 50
3. 879. 985 = 29. 6645≈ 29. 66
4. 50. 01 = 7 . 07177≈ 7 . 1
Practice 9.1.31. − 49 = - 7
2. − 81 The radicand is negative sothis is not defined using realnumbers.
3. − 1 = - 1
4. − − 121 The radicand is negative so this is not defined using realnumbers.
Practice 9.1.41. 5 = 6 49 = 5 + 6•7 = 5 +42 = 47
2. 7 - 5 25 - 9 = 7 - 5 16 = 7-5•4 = 7- 20 = -13
3. 8 - 12 − 4 The radicand is
negative so this is not defined usingreal numbers.
4. − 25 + 121 = - 5 + 11 = 6
5. − 16 − 81 = - 4 - 9 = - 13
6. 8 - 4 • 16 − 25 • 36 = 8 - 4•4 - 5•6 = 8 - 16 - 30 = -38
7. 12 - 2• 9 + − 49 • 4 The radicand is negative so this is not defined using realnumbers.
8. (5 + 81 )(3 - 121 ) = (5 + 9)(3 - 11) = 14(-8) = -112
9. (2 - 16 )(4 - − 64 )The radicand is negative so this is not defined using realnumbers.
Practice 9.1.51. a = 9.1, b = 1.3
c = a 2 + b 2
c = ( 9 . 1 ) 2 + ( 1 . 3 ) 2
c = 82. 81 + 1 . 69c = 84. 5 c = 9.2
2. a = 408, b = 1700c = 4082 + 17002
c = 3 , 056, 464c = 1748.3 = 1750 rounded
3. s = 2 gdg = 32, d = 36s = 2 • 32• 36s = 2304s = 48
4. g = 32 and d = 49s = 2 • 32• 49s= 3136s = 56
5. g = 9.8 and d = 15s = 2 • 9 . 8 • 15s = 294
254 CHAPTER 9 RADICAL EXPRESSIONS AND EQUATIONS
s = 17
Exercise Set 9.11. 82 = 64 so 64 = 8
3. (9 11
)2 = (81121
) so 81121
= 9 11
5. (0.2)2 = 0.04 so 0 . 04 = 0 . 2
7. (1.1)2 = 1.21 so 1 . 21 = 1 . 1
9. 15. 4 = 3 . 924 ≈ 4
11. 254. 82 = 15. 963 ≈16. 0
13. 0 . 05 = 0 . 2236 ≈ 0 . 22
15. − 25 The radicand is negative sothis is not defined for real numbers.
17. − 169 = - 13
19. − − 4 The radicand is negative sothis is not defined for real numbers.
21. 15 + 8 64 = 15 + 8•8 = 15 + 64 = 79
23. 36 - 2 49 = 36 -2•7 = 36 - 14 = 22
25. − 36 + 64 =-6 +8 = 2
27. 3 4 - 5 16 = 3•2 - 5•4 = 6 - 20 = -14
29. 8 + 4 9 + 36 • 49 = 8+4•3 +6•7 = 8 + 12 + 42 = 62
31. (12 - 100 ) ( 4 + 25 ) = (12 - 10)(4 + 5) = 2• 9 = 18
33. ( 4 + 2 4 ) ( 5 - 3 25 ) =(4 + 2•2) (5 - 3•5) = (4 + 4) (5 - 15) = 8(-10) = -80
35. c = a 2 + b 2
a = 15, b = 18c = 152 + 182
c = 549
c =23
37. a = 0.0013, b = 0.056c = 0 . 00132 + 0 . 0562
c = 0 . 0031377c = 0.056
39. a = 735, b = 5010c = 7352 + 50102
c = 25, 640, 325c = 5060
41. a = 105.050, b = 60.002c = ( 105. 050) 2 + ( 60. 002) 2
c = 14635. 743c = 120.98
43. s = 2 5 d d = 405 feets = 2 5 • 405s = 2 2025s = 2•45 = 90 mph
45. d = 80 feets = 2 5 • 80s = 2 400s = 2•20 = 40 mph
47. 117 = 3•3•13
48.
-10-9-8-7-6-5-4-3-2-10123456789
10
y
-10-9-8-7-6-5-4-3-2-101 2 3 4 5 6 7 8 910 x
Data #7
49. a(a2 - ab + b2) +b(a2 - ab + b2) =a3 - a2b + ab2 + a2b - ab2 + b3 = a3 + b3
255CHAPTER 9 RADICAL EXPRESSIONS AND EQUATIONS
50.125h 11 − 75h 9
− 5 h 7 = 125h 11
− 5 h 7 − 75h 9
− 5 h 7 =
-25h4 +15h2
51.
3 x 2 + 3 x + 3 + 5
2 x - 2
2 x - 2 6 x 3 + 0 x 2 + 0 x - 1 6 x 3 − 6 x 2 Subtract
6 x 2 + 0 x 6 x 2 - 6 x Subtract
6 x - 1 6 x - 6 Subtract
5
52. ( ( 2 a ) ( 3 ab2 )
30a 3 ) 3 = ( 6 a 2 b 2
30a 3 ) 3 = ( b 2
5 a ) 3
=b 6
125a 3
53. a2b3ab-2 = a3b
54. ( t 3 v
w − 1 ) (
t − 4
w 3 ) = t
− 1 v
w 2 = v
tw2
55.
x 4
y 2
x 3
y 5
= x 4
y 2 •
y 5
x 3 = xy3
56.2 m − 3
( 2 m ) − 3 •
3 m − 2
( 3 m ) − 2 =
2 ( 2 m ) 3
m 3 •
3 ( 3 m ) 2
m 2 =
2 • 8 m 3
m 3 •
3 • 9 m 2
m 2 = 16• 27 = 432
57. (3s +2t)(3s - 2t) = (3s)2 - (2t)2 =
9s2 - 4t2
58.18x 3
9 y 2 •
y 5
2 x = 2 • 9 x 3
9 y 2 •
y 5
2 x = x 2 y 3
59. 1 - 1 5 x
+ 3 2 x
=
1 • 10x 10x
- 1 5 x
• 2 2
+ 3 2 x
• 5 5
=
10x - 2 + 1510x
= 10x + 1310x
60. 45 - 14x + x2 = x2 - 14x + 45 = (x - 9)(x - 5)
61. 12x2 - 30x + 12 = 6(2x2 - 5x + 2) =6(2x - 1)(x - 2)
62. x2 + 6x + 9 = (x+3)(x+3) = (x +3)2
63. 10x + 10x2 - 75x - 35 = 10x2 - 65x - 35 = 5(2x2 - 13x - 7)5(2x + 1)(x - 7)
64. 6(x -1) = 2 3
(x-9)
3•6(x-1) = 3•2 3
(x-9)
18x - 18 = 2(x - 9)18x - 18 = 2x - 1816x = 0x = 0
65. x2 - 64 = 0(x +8)(x - 8) = 0x + 8 = 0 or x - 8 = 0x = -8 or x = 8
66. x2 - 9 = (x+3)(x-3) = LCD2
x - 3 − 3
x + 3 = 12
x 2 − 9 2
x - 3 − 3
x + 3 = 12
( x + 3 ) ( x - 3 )
( x + 3 ) ( x - 3 ) 2
x - 3 − ( x + 3 ) ( x - 3 )
3 x + 3
=
( x + 3 ) ( x - 3 ) 12
( x + 3 ) ( x - 3 ) 2(x+3)-3(x-3) = 122x + 6 - 3x + 9 = 12-x + 15 = 12-x = -3
256 CHAPTER 9 RADICAL EXPRESSIONS AND EQUATIONS
x = 32
3 - 3 − 3
3 + 3 = 12
3 2 − 9 2/0 - 3/6 = 12/0 Undefined3 is a restricted value. There is nosolution.
67. Eq 1: 3x - 11y = -5Eq 2: 7x + 2y = 162 times Eq 1: 2(3x - 11y) = 2(-5)11 times Eq 2: 11(7x + 2y) = 11•16Eq 1 transformed: 6x - 22y = -10 Eq 2 trAnsformed: 77x +22y=176Add together 83x = 166x = 2Substitute to solve for y:3•2 - 11y = -56 - 11 y = -5-11y = -11y = 1The solution is the ordered pair(2,1).
68. i = prti = 17000•0.065•3 = 3315Total = $17,000 + $3,315 =$20,315
69.48 right
65 questions= x right
38 questions65x = 38•4865x = 1824x = 28.06 or 28 questions
70. Let x = number of years since 1928y = current gold medal distance y = 39.4 + 0.51x
257CHAPTER 9 RADICAL EXPRESSIONS AND EQUATIONS
Section 9.2 Simplifying Radical Expressions
Practice 9.2.11. 121w = 121 w = 11 w
2. 81xy = 81 xy = 9 xy
3.16
t 2 = 16
t 2 = 4
t
4.25y 2 t
z 2 = 25y 2 t
z 2 = 5 y t
z
5. ( 3 a ) 2 = 3a if a ≥ 0
Practice 9.2.21. 2 • 2 d = 4 d = 4 d = 2 d
2. 18 • 8 b = 144b = 12 b
3.5 w h
w = 5 wh
w = 5 h
4.48w g
3 w = 48wg
3 w = 16g =
16 g = 4 g
Practice 9.2.31. 8 = 4 • 2 = 4 • 2 = 2 2
2. 75 = 25• 3 = 25 • 3 = 5 3
3. 128 = 64• 2 = 64 • 2 = 8 2
4.2449
= 24
49= 4 • 6
7 =
4 • 6 7
= 2 6 7
Practice 9.2.41. 5000 = 5 4 • 2 3 = 5 2 2 2 =
50 2
2. 432 = 2 • 2 • 2 • 2 • 3 • 3 • 3 = 2 4 • 3 3 = 2 2 • 3 3 = 12 3
3. 392 = 2 • 2 • 2 • 7 • 7 = 2 3 • 7 2 = 2•7 2 14 2
4. 968 = 2 • 2 • 2 • 11• 11 = 2 3 • 112 = 2 • 11 2 = 22 2
Practice 9.2.51. d 36 = d 18 since 36÷2 = 18
2. x 13 = x 6 x since 13÷2 = 6 with remainder 1.
3. y 81m 32 = y 40m 16 y since 81÷2=40 with remainder 1 and 32÷2 =16
4. x 5 y 12z 9 = x 2 y 6 z 4 xz since 5÷2= 2 with remainder 1, 12÷2 = 6and 9÷2 = 4 with remainder 1.
5. m 49n 4 r 25 = m 24n 2 r 12 mr since49÷2=24 with remainder 1, 4÷2 = 2and 25÷2 = 12 with remainder 1.
258 CHAPTER 9 RADICAL EXPRESSIONS AND EQUATIONS
Practice 9.2.61. 48a 4 b 6 = 2 4 • 3 a 4 b 6 =
2 2 a 2 b 3 3 = 4 a 2 b 3 3
2. 75x 8 y 10 = 3 • 5 2 x 8 y 10 = 5 x 4 y 5 3
3. 128r 15s 11t = 2 7 r 15s 11t = 2 3 r 7 s 5 2 rst = 8 r 7 s 5 2 rst
4. 44x 17y 13z 8 = 2 2 • 11x 17y 13z 8 = 2 x 8 y 6 z 4 11xy
5. x 2 − 4 x + 4 = ( x - 2 ) 2 = x - 2
6. x 2 + 12x + 36 = ( x + 6 ) 2 = x + 6
Practice 9.2.71. 7 m 49m 7 = 7 m 7 2 m 7 =
7 m•7 m 3 m = 49m 4 m
2. 2 b 48b 25 = 2 b• 2 4 • 3 b 25 = 2 b•2 2 b 12 3 b = 8 b 13 3 b
3.6 x 4 72x 3 y 5
8 y = 3 x 4 2 • 6 2 x 3 y 5
4 y =
3 x 4 • 6 xy2 2 xy4 y
= 9 x 5 y 2 xy2
4.2 m 45m 7 n 4
3 n 4 = 2 m 3 2 • 5 m 7 n 4
3 n 4 =
2 m•3 m 3 n 2 5 m
3 n 4 = 2 m 4 5 m
n 2
Exercise Set 9.21. 49m = 49 m = 7 m
3.s 4
= s
4 = s
2
5.10
y 2 = 10
y 2 = 10
y
7. ( 5 y ) 2 = 5 y
9. 4 x 2 = 4 x 2 = 2 x
11. 169x 4 = 169 x 4 = 13x 2
13. 7 • 7 t = 7 2 t = 7 t
15.44 rw
11w = 44rw
11w = 4 r = 2 r
17. 27 = 9 3 = 3 3
19. 50 = 25 2 = 5 2
21. 200 = 100 2 = 10 2
23.409
= 40
9 = 4 10
9 = 2 10
3
25.1225
= 12
25= 4 3
25= 2 3
5
27. 150 = 2 • 3 • 5 2 = 5 6
29. 5832 = 2 3 • 3 6 = 2 • 3 3 2 = 54 2
31. x 15 = x 7 x
33. a 17 = a 8 a
35. x 7 y 9 z 3 = x 3 y 4 z xyz
37. r 42s 34t 16 = r 21s 17t 8
39. 12y 5 = 4 • 3 y 5 = 2 y 2 3 y
41. 50a 9 b 15 = 5 2 • 2 a 9 b 15 = 5 a 4 b 7 2 ab
43. 100w 25t 16 = 102 w 25t 16 = 10w 12t 8 w
45. 80r 2 s 5 t 3 = 2 4 • 5 r 2 s 5 t 3 = 2 2 rs2 t 5 st = 4 rs2 t 5 st
259CHAPTER 9 RADICAL EXPRESSIONS AND EQUATIONS
47. 60x 20y 12z 13 = 2 2 • 3 • 5 x 20y 12z 13 =
2 x 10y 6 z 6 15z
49. x 2 + 16x + 64 = ( x + 8 ) 2 = | x + 8 |
51. x 2 − 8 x + 16 = ( x - 4 ) 2 = | x - 4 |
53. 4 x 2 + 20x + 25 = ( 2 x + 5 ) 2 = |2 x + 5 |
55. 4 x 25x 10 = 4 x•5 x 5 = 20x 6
57. 8 s 36s 11 = 8 s•6 s 5 s = 48s 6 s
59. 9 wx 144w 9 x 4 = 9 wx•12w 4 x 2 w = 108w 5 x 3 w
61. 4 x 3 y 4 121x 11y 16 = 4 x 3 y 4 • 11x 5 y 8 x = 44x 8 y 12 x
63.10a 3 27a 36b 10
15b = 2 a 3 3 3 a 36b 10
3 b =
2 a 3 • 3 a 18b 5 3 3 b
= 2 a 21b 4 3
65.8 x 50x 6 y 7
5 y = 8 x 25• 2 x 6 y 7
5 y =
8 x•5 x 3 y 3 2 y 5 y
= 8 x 4 y 2 2 y
67. Let r = speed at long play2r = speed at standarddistance = rate• time1(r) + 1.5(2r) = 246r + 3r = 2464r = 246r = 61.5 meters per hour
68. Let t = time of slow traint - 1 = time of faster trainDistances are equal.distance = rate •time48t + 64(t -1) = 24448t + 64t - 64 = 244112t = 308t = 2.75 hours10am + 2.75 hrs = 12:45 pm
69. 2y = 6-4xy = 3 - 2xm = -2 and the y intercept is (0,3).
70. 81x2 - 1 = (9x+1)(9x-1)
71. (0,-3) and (3,0)
-10-9-8-7-6-5-4-3-2-10123456789
10
y
-10-9-8-7-6-5-4-3-2-101 2 3 4 5 6 7 8 910 x
Data #8
72.6 x - 30
30x 2 − 15x - 675= 6 ( x - 5 )
15( 2 x 2 − x - 45) =
6 ( x - 5 ) 15( 2 x + 9 ) ( x - 5 )
= 2 5 ( 2 x + 9 )
73.
-10-9-8-7-6-5-4-3-2-10123456789
10
y
-10-9-8-7-6-5-4-3-2-101 2 3 4 5 6 7 8 910 x
Data #9
74.
260 CHAPTER 9 RADICAL EXPRESSIONS AND EQUATIONS
-8-7-6-5-4-3-2-10123456789
10
y
-10-9-8-7-6-5-4-3-2-101 2 3 4 5 6 7 8 910 x
Data #10
75.5 6
x - 1 2
≥ 2 x - 4
6 • 5 6
x - 6 • 1 2
≥ 6 • 2 x - 6 • 4
5x - 3 ≥12x - 24-7x ≥ - 21x ≤ 3
76. 5x2 - 45 = 05(x2 - 9) = 05(x+3)(x-3) = 0x + 3 = 0 or x - 3 = 0x = -3 or x = 3
77. LCD = x(x + 2)2
x + 2 + 1
x = 5
x + 2
x ( x + 2 ) 2
x + 2 + x ( x + 2 )
1 x
= x ( x + 2 ) 5
x + 2 2x + x + 2 = 5x3x + 2 = 5x2 = 2xx = 1
78. Eq 1: x + y = 3Eq 2: 5x = 15 - 5yEq 2 divided by 5: x = 3 - ySubstitute Eq 2 in Eq 1:3-y + y = 33 = 3This is an identity. There are aninfinite number of solutions.Solve for y: y = 3-xThe solutions may be written (x,3-x).
79. Eq 1: 4x + 7y = -5
Eq 2: 6x + 8y = 103 times Eq 1: 3(4x + 7y) = 3(-5)-2 times Eq 2: -2(6x + 8y) = -2(10)Eq 1 transformed: 12x + 21y = -15Eq 2 transformed: -12x - 16y = -20Add together 5y = -35y = -7Substitute to solve for x: 4x+7(-7) = -54x - 49 = -54x = 44x = 11The solution is the ordered pair (11,-7).
80. -(3x+4) -(8-2x) = -3x - 4 - 8 + 2x =-x - 12
81.1812
÷ 4530
= 1812
• 3045
=
2 • 3 • 3 2 • 2 • 3
• 2 • 3 • 5 3 • 3 • 5
= 1
82. -m-1(-m3)(-2m-2) = 2
83. ( 2 a 3 b
) 2 ( a
2 b − 1 ) 3 (
a 2
b ) − 1 =
4 a 2
9 b 2 •
a 3
8 b − 3 •
a - 2
b − 1 = a 3
18b − 2 = a 3 b 2
18
84. 3.4 x 104 + 5.2 + 103 = 34000+5200 = 39200 = 3.92 x 104
85. (x - 15p)(x + 15p) = x2 - 225p2
86. ( 3 5
m 3 n + 2 ) + (
125
m 3 n - mn+ 2 ) +
( 6 mn- 2 ) =
= 155
m 3 n + 5 mn+ 2 = 3 m 3 n + 5 mn+ 2
87. 12x2 + 44x + 40 = 4(3x2 + 11x + 10) =4(3x + 5)(x + 2)
88. 48x3 - 104x2 + 48x = 8x(6x2 - 13x + 6) = 8x(3x-2)(2x-3)
89. a)Let x = number of years since
261CHAPTER 9 RADICAL EXPRESSIONS AND EQUATIONS
1932y = the winning timey = 79.4 - 0.33xb) 2008 - 1932 = 76 = xy = 79.4 -0.33(76) y = 54.32 secondsc)79.4 - 0.33x <6019.4 <0.33xx >58.8x > 59 years after 1932Occurred in 1991
Section 9.3 Multiplying and Dividing Radicals
Practice 9.3.11. 32 12 = 2 5 • 2 2 • 3 = 2 7 • 3 =
2 3 2 • 3 = 8 6
2. 75 27 = 3 • 5 2 • 3 3 = 3 4 • 5 2 = 32•5 = 9•5 = 45
3.15x 7 y
35x
y 3 = 3 • 5 x•5 • 7 x
7 y 4 =
3 • 5 2 • 7 x 2
7 y 4 = 5 x 3
y 2
4. 8 x 2 y 3 12x 4 y 5 = 2 3 • 2 2 • 3 x 2 y 3 x 4 y 5 = 2 5 • 3 x 6 y 8 = 2 2 x 3 y 4 3 = 4 x 3 y 4 3
5. 15m 11n 7 75m 5 n 12 = 3 • 5 m 11n 7 • 3 • 5 2 m 5 n 12 = 3 2 • 5 3 m 16n 19 = 3 • 5 m 8 n 9 5 n
15m 8 n 9 5 n
Practice 9.3.2
1.3
11= 3
11•
11
11= 3 11
( 11) 2 =
3 1111
2.7
12= 7
12•
3
3 = 7 3
36=
7 3 6
3.10
5 •
5
5 = 10 5
( 5 ) 2 •
10 5 5
=
2 5
4.11
7 a = 11
7 a •
7 a
7 a = 77a
( 7 a ) 2 =
77a 7 a
5.4 m 2
6 m = 4 m 2
6 m •
6 m
6 m =
4 m 2 6 m
( 6 m ) 2 = 4 m 2 6 m
6 m =
2 m 6 m 3
262 CHAPTER 9 RADICAL EXPRESSIONS AND EQUATIONS
6.12y 2
8 y = 12y 2
8 y •
2 y
2 y =
12y 2 2 y
16y 2 = 12y 2 2 y
4 y = 3 y 2 y
Exercise Set 9.31. 48 36 = 2 4 • 3 2 2 • 3 2 =
2 2 • 2 • 3 3 = 24 3
3. 10 5 = 2 • 5 • 5 = 2 • 5 2 = 5 2
5. 20x 10x 9 = 2 • 10x•10x 9 = 2 • 102 x 10 = 10x 5 2
7.8 t
q 3
24t 4
q = 8 t•8 • 3 t 4
q 4 =
8 2 • 3 t 5
q 4 = 8 t 2 3 t
q 2
9. 12m 45m 11 = 2 2 • 3 • 3 2 • 5 m 12 = 2 • 3 m 6 3 • 5 = 6 m 6 15
11. 54x 2 y 3 27x 10y 12 = 2 • 3 3 • 3 3 x 12y 15 = 2 • 3 6 x 12y 15
= 3 3 x 6 y 7 2 y = 27x 6 y 7 2 y
13. 54x 3 y 5 12x 3 y 4 = 2 • 3 3 2 2 • 3 x 6 y 9 = 2 3 • 3 4 x 6 y 9 =
2 • 3 2 x 3 y 4 2 y = 18x 3 y 4 2 y
15.4
6 = 4
6 •
6
6 = 4 6
6 = 2 6
3
17.6
13= 6
13•
13
13= 6 13
13
19.8
2 = 8
2 •
2
2 = 8 2
( 2 ) 2 =
8 2 2
= 4 2
21.7
6 m = 7
6 m •
6 m
6 m =
42m
( 6 m ) 2 = 42m
6 m
23.3
5 t = 3
5 t •
5 t
5 t = 15t
( 5 t ) 2 = 15t
5 t
25.5 x 3
10x = 5 x 3
10x •
10x
10x =
5 x 3 10x
( 10x ) 2 = 5 x 3 10x
10x = x 2 10x
2
27.7 d 2
11d = 7 d 2
11d •
11d
11d =
7 d 2 11d
( 11d ) 2 = 7 d 2 11d
11d = 7 d 11d
11
29. 3(x-xy) -y2 = 3x - 3xy - y2
3(-2) - 3(-2)(-3) - (-3)2 = -6 - 18 - 9 = - 33
30. Eq 1: 9x + y = 4Eq 2: 3x - 13 = 2ySolve Eq 1 for y: y = 4 - 9xSubstitute in Eq 2:3x - 13 = 2(4 - 9x) 3x - 13 = 8 - 18x21x = 21x = 1Substitute to solve for y:9(1) + y = 49 + y = 4y = -5The solution is the ordered pair(1,-5).
31. Let t = time of freight traint - 3/4 = time of passenger traindistance = rate • timeThe distances are equal.40t = 45(t - 3/4)40t = 45t - 135/4-5t = -135/4-20t = -135t = 6.75 hours8:15 + 6 hr and 45 min = 3 pmNo, it will not have caught up.
32. amount = base • rateLet x = percent of copper in 35 gram
263CHAPTER 9 RADICAL EXPRESSIONS AND EQUATIONS
bar. 40(0.60) + 35x = (40+35)0.5324+35x = 75•0.5324+35x = 39.7535x = 15.75x = 0.45 or 45%
33. d = 1/2 n(n - 3)14 = 1/2 (n2 - 3n)28 = n2 - 3nn2 - 3n - 28 = 0(n - 7)(n + 4) = 0n - 7 = 0 or n + 4 = 0n = 7 sides or n = -4 disregard
We can’t have a -4 sides.
34. f(x) = 2x2 - 3x + 9f(-1) = 2(-1)2 - 3(-1) + 9f(-1) = 2 + 3 + 9 = 14
35. 3 w 3 − 4 w 2 + w + 4 - 3
w + 2
w + 2 3 w 4 + 2 w 3 − 7 w 2 + 6 w + 5 3 w 4 + 6 w 3 Subtract
− 4 w 3 − 7 w 2 − 4 w 3 − 8 w 2 Subtract
w 2 + 6 w w 2 + 2 w Subtract
4 w + 5 4 w + 8 Subtract
- 3
36. (10z-11y)(10z+11y) = 100z2 -121y2
37. -p( 5 t p
) 2 ( 2 p 5 t
) = - p ( 25t 2
p 2 ) (
2 p 5 t
) =
- 10t
38. (3xy)3(-2xy)-2=33x3y3(-2)-2x-2y-2 = 33(-2)-2xy =27xy
4
39. (4d-5)(3d +7) = 12d2 +28d - 15d - 3512d2 + 13d - 35
40. ( − 4 m 2
n − 2 ) (
− 4 m − 2
n 3 ) − 2 (
m − 4
n ) =
( − 4 m 2 n 2
1 ) (
( − 4 ) − 2 m 4
n − 6 ) (
m − 4
n ) =
( − 4 m 2 n 2
1 ) (
m 4 n 6
( - 4 ) 2 ) (
m − 4
n ) =
( − 4 m 2 n 2
1 ) (
m 4 n 6
16) (
m − 4
n ) =
− m 2 n 7
4
41. ( b − 2
− 2 b ) − 3 = b 6
( - 2 ) − 3 b − 3 =
( - 2 ) 3 b 6 b 3 = - 8 b 9
42.3
m − 1
5
4
m
= 5 m ( 3
m − 1
5 )
5 m ( 4
m )
=
5 • 3 - m 5 • 4
= 15 - m 20
43. 54m 11n 3 = 2 • 3 3 m 11n 3 = 3 m 5 n 2 • 3 mn = 3 m 5 n 6 mn
44. 8x2 - 20x - 12 = 4(2x2 - 5x - 3)=4(2x + 1)(x - 3)
45. 9x2 - 16 = (3x + 4)(3x - 4)
46. 2x3 + 32x2 + 6x3 - 40x =8x3 + 32x2 - 40x = 8x(x2 + 4x - 5)= 8x(x + 5)(x - 1)
47. x2 - 3x - 10 = 0(x - 5)(x + 2) = 0x - 5 = 0 or x + 2 = 0x = 5 or x = -2
48. 12x2 + 15x = -312x2 + 15x + 3 = 03(4x2 + 5x + 1) = 03(4x + 1)(x + 1) = 04x + 1 = 0 or x + 1 = 04x = -1 or x = -1x = -1/4 or x = -1
49. x2 - 5x - 14 = (x - 7)(x + 2) = LCD27
x 2 − 5 x - 14+ 2
x + 2 = 3
x - 7
264 CHAPTER 9 RADICAL EXPRESSIONS AND EQUATIONS
27( x + 2 ) ( x - 7 )
+ 2 x + 2
= 3 x - 7
( x + 2 ) ( x - 7 ) 27
( x + 2 ) ( x - 7 ) + ( x + 2 ) ( x - 7 )
2 x + 2
=
( x + 2 ) ( x - 7 ) 3
x - 7 27 + 2(x -7) = 2(x + 2)27 + 2x - 14 = 2x + 413 + 2x = 2x + 413 = 4This is a contradiction. There is nosolution.
50. a) 80 years after 1869 (1949) therewere 30.7 million students enrolled.b)This represents the number ofstudents enrolled 100 years after
1869 (1969).c) t years after 1869 there were 40million students enrolled.d)The number of students enrolled tyears after 1869 is 4 times the
number of students enrolled in1869.
Section 9.4 Adding and Subtracting Radical Expressions
Practice 9.4.11. 12 3 + 8 3 = 20 3
2. 5 3 + 2 5 + 6 3 + - 5 = 11 3 + 5
3. 12 7 + 4 11 − 4 7 + 6 11 = 8 7 + 10 11
4. 22 5 y − 8 y 5 y This cannot besimplified any further.
5. 3 t − 8 3 t = - 7 3 t
6. 12 6 y + 8 6 y − 7 6 y = 13 6 y
7. 7 m + 15 7 m 5
= 7 m + 3 7 m =
4 7 m
8. 3 8 s − 8 3 s 6
= 3 8 s − 4 3 s 3
Practice 9.4.21. 7 75 + 12 = 7 3 • 5 2 + 2 2 • 3 =
7 • 5 3 + 2 3 = 35 3 + 2 3 =37 3
2. 8 50 − 8 = 8 2 • 5 2 − 2 3 = 8 • 5 2 − 2 2 = 40 2 - 2 2 = 38 2
3. 4 12 + 50 − 75 + 2 8 = 4 3 • 2 2 + 2 • 5 2 − 3 • 5 2 + 2 2 3 = 4 • 2 3 + 5 2 − 5 3 + 2 • 2 2 = 8 3 + 5 2 − 5 3 + 4 2 = 3 3 + 9 2
4. 3 27 + 32 − 12 + 4 50 =
265CHAPTER 9 RADICAL EXPRESSIONS AND EQUATIONS
3 3 3 + 2 5 − 2 2 • 3 + 4 2 • 5 2 = 3 • 3 3 + 2 2 2 − 2 3 + 4 • 5 2 = 9 3 + 4 2 − 2 3 + 20 2 = 7 3 + 24 2
5. 4 m 50m + 6 8 m 3 = 4 m 2 • 5 2 m + 6 2 3 m 3 = 4 m•5 2 m + 6 • 2 m 2 m = 20m 2 m + 12m 2 m = 32m 2 m
6. 2 49t + 64t − t = 2 • 7 t + 8 t − t = 14 t + 8 t − t = 21 t
Practice 9.4.31. 5 ( 20 + 2 ) + 45 =
100 + 2 5 + 9 • 5 = 10 + 2 5 + 3 5 = 10 + 5 5
2. 7 y ( y + 4 ) - 6 y = 7 y 2 + 28 y − 6 y = 7 y + 22 y
3. 3 t ( t + 3 ) - 4 t = 3 t 2 + 9 t − 4 t = 3 t + 5 t
4. ( 3 x + 3 ) 2 + 2 3 x = ( 3 x ) 2 + 2 • 3 • 3 x + ( 3 ) 2 + 2 3 x = 3 x + 6 3 x + 9 + 2 3 x =3 x + 8 3 x + 9
5. 5 3 + 1
3 = 5 3 + 1
3 •
3
3 =
5 3 + 3
( 3 ) 2 = 5 3 + 3
3 =
15 3 3
+ 3 3
= 16 3 3
6.2
5 − 5 = 2
5 •
5
5 − 5 =
2 5 5
− 5 = 2 5 5
− 5 5 5
=
− 3 5 5
7.1
3 + 3
7 − 7 =
1
3 •
3
3 + 3
7 •
7
7 − 7 =
3 3
+ 3 7 7
− 7 7 7
= 3 3
- 4 7 7
Exercise Set 9.4
1. 6 7 + 15 7 = 21 7 3. 5 3 + 2 3 − 8 3 = - 3
5. 8 5 s + 9 5 s − 3 5 s = 14 5 s
7. 3 5 + 2 2 x + 7 5 − 5 2 x = 10 5 − 3 2 x
9. 6 a − 4 6 a 2
= 6 a − 2 6 a = - 6 a
11. 11 5 c − 12 5 c 4
=
11 5 c − 3 5 c = 8 5 c
13. 3 5 n + 18 5 n 3
= 3 5 n + 6 5 n =
9 5 n
15. 2 t - 2 w This cannot besimplified.
17.5 2
2 s − 2 s = 5 2
2 s − 2 2 s 2
=
3 2 s 2
19. 7 8 + 32 = 7 2 3 + 2 5 = 7 • 2 2 + 2 2 2 = 14 2 + 4 2 = 18 2
21. 4 8 − 6 50 = 4 2 3 − 6 2 • 5 2 = 4 • 2 2 − 6 • 5 2 = 8 2 − 30 2 = - 22 2
23. 4 36y + 2 49y − 8 25y =
266 CHAPTER 9 RADICAL EXPRESSIONS AND EQUATIONS
4 • 6 y + 2 • 7 y − 8 • 5 y = 24 y + 14 y − 40 y = - 2 y
25. 5 12 − 3 50 + 2 75 − 6 8 = 5 4 • 3 − 3 25• 2 + 2 25• 3 − 6 4 • 2 = 5 • 2 3 − 3 • 5 2 + 2 • 5 3 − 6 • 2 2 = 10 3 − 15 2 + 10 3 − 12 2 = 20 3 − 27 2
27. 5 x 12x + 4 75x 3 = 5 x 4 • 3 x + 4 25• 3 x 3 = 5 x•2 3 x + 4 • 5 x 3 x = 10x 3 x + 20x 3 x = 30x 3 x
29. 3 r 24r − 2 54r 3 − 2 r 6 r = 3 r 4 • 6 r − 2 9 • 6 r 3 − 2 r 6 r = = 3 r•2 6 r − 2 • 3 r 6 r − 2 r 6 r = 6 r 6 r − 6 r 6 r − 2 r 6 r = - 2 r 6 r
31. 2 ( 8 + 2 ) + 3 32 = 16 + 2 2 + 3 16• 2 =
4 + 2 2 + 3 • 4 2 = 4 + 2 2 + 12 2 = 4 + 14 2
33. 6 ( 6 + 3 ) + 28 = 36 + 3 6 + 4 • 7 =
6 + 3 6 + 2 7
35. 3 ( 12 − 6 ) + 27 = 36 − 6 3 + 9 • 3 =
6 - 6 3 + 3 3 = 6 - 3 3
37. 6 w ( w + 4 ) + 36w = 6 w 2 + 24 w + 6 w = 6 w + 30 w
39. ( 5 a − 3 ) 2 + 2 5 a = ( 5 a ) 2 − 2 • 3 5 a + 9 + 2 5 a = 5 a − 6 5 a + 9 + 2 5 a = 5 a - 4 5 a + 9
41. ( 2 w − 6 ) 2 − 3 w 2 = ( 2 w ) 2 − 2 • 6 2 w + ( - 6 ) 2 − 3 w 2 = 2 w - 12 2 w + 36 - 3 w 2
43. 2 5 + 1
5 = 2 5 + 1
5 •
5
5 =
2 5 + 5 5
= 2 5 ( 5 5
) + 5 5
=
10 5 + 5 5
= 11 5 5
45. 4 3 − 2
3 = 4 3 − 2
3 •
3
3 =
4 3 − 2 3 3
= 4 3 ( 3 3
) − 2 3 3
=
12 3 − 2 3 3
= 10 3 3
47.1
8 + 1
12=
1
8 •
2
2 + 1
12•
3
3 =
2
16+ 3
36= 2
4 + 3
6
49.2
5 + 5
6 − 5 =
2
5 •
5
5 + 5
6 •
6
6 − 5 =
2 5 5
+ 5 6 6
− 5 =
2 5 5
+ 5 6 6
− 5 ( 5 5
) =
2 5 − 5 5 5
+ 5 6 6
=
- 3 5 5
+ 5 6 6
51. The base is 5 and the exponent is 2.-52 = -25
52. Let x = the first integer; x +1 =second integer; x + 2 = thirdinteger
x + (x + 1) + (x + 2) = 2(x + 2) +9x + x+ 1 +x +2 = 2x + 4 + 93x + 3 = 2x + 13x = 10x + 1 = 10 + 1 = 11x + 2 = 10 + 2 = 12
267CHAPTER 9 RADICAL EXPRESSIONS AND EQUATIONS
53. y = 5x - 1 m = 5 Slope of a perpendicular line is -1/5.y = mx + b0 = -1/5(0) + bb = 0y = -1/5x
54. Percent for math and computersciences =241,000,000/4,192,000,000 x100%=5.7%Total in 1990 = 9,679,000,000Percent for math and computersciences =841,000,000/8,838,000,000x 100%=9.5%The percent for math increased.
55. -12x2 + 14x + 6 = -2(6x2 -7x - 3)-2(3x + 1)(2x - 3)
56.15x - 15
10x 2 − 10= 15( x - 1 )
10( x 2 − 1 ) =
3 ( x − 1 ) 2 ( x + 1 ) ( x - 1 )
= 3 2 ( x + 1 )
57. 8(25 -2•32) = 8(25 - 2•9) = 8(25-18)=
8(7) = 56
58. 1 - 1 6 x
+ 5 2 x
=
1 ( 6 x 6 x
) - 1 6 x
+ 5 2 x
( 3 3
) =
6 x - 1 + 156 x
= 6 x + 146 x
= 2 ( 3 x + 7 ) 2 • 3 x
=
3 x + 7 3 x
59.x 2 − 4 x - 3
• x 2 − x - 6
x 2 + 4 x + 4 =
( x + 2 ) ( x - 2 ) x - 3
• ( x - 3 ) ( x + 2 ) ( x + 2 ) ( x + 2 )
= x - 2
60.648x 2 y 2
9 x 3 = 2 3 • 3 4 x 2 y 2
9 x 3 =
2 • 3 2 xy 2
9 x 3 = 2 • 9 xy 2
9 x 3 = 2 y 2
x 2
61. ( 64 − 9 ) 2 = ( 8 - 3 ) 2 = ( 5 ) 2 = 25
62. 12a 9 b 2 = 4 • 3 a ( 3 b ) =( 3 b ) • 2 3 a = 6 b 3 a
63. -(f-2g-3h)-(6h-f)-(-2g+3h)=-f+2g+3h-6h+f+2g-3h =4g - 6h
64. 0 = x2 - 11x + 240 = (x - 8)(x - 3)x - 8 = 0 or x - 3 = 0x = 8 or x = 3
65. -30 -61x = 30x2
30x2 + 61x + 30 = 0ac = 30•30 = 900Product Sum20(45) 6525(36) 6130x2 + 25x + 36x + 30 = 05x(6x + 5) + 6(6x + 5) = 0(5x + 6)(6x + 5) = 05x + 6 = 0 or 6x + 5 = 05x = -6 or 6x = -5x = -6/5 or x = -5/6
66. 0.15(12-x)+0.25x=5+0.1x1.8 -0.15x + 0.25x = 5 +0.1x1.8 +0.10x = 5+0.1x1.8 = 5This is a contradiction. There is nosolution.
67.7 m
= 3 4
− m - 5 m
4 m•7 m
= 4 m•3 4
− 4 m•m - 5
m 28 = 3m -4(m-5)28 = 3m - 4m + 208 = -mm = -8
68. The domain is -7 ≤x ≤6. The rangeis
2 ≤ y≤ 7.
69. a)E = 70.3 +0.16tb)E(60) = 70.3 +0.16(60) =79.9By 2020 (1960 + 60) the average lifeexpectancy will be about 79.9 years.c)60 = 70.3 +0.16t-10.3 =0.16t
268 CHAPTER 9 RADICAL EXPRESSIONS AND EQUATIONS
t = -64.375 years This means in about 1896 (1960 -
64) the life expectancy was 60 years.d)E(0) = 70.32(70.3) = 140.6140.6 =70.3 +0.16t70.3 = 0.16tt = 439.375 yearsIn 2399 the average life expectancywill be twice the average lifeexpectancy in 1960.
Section 9.5 Solving Equations that Involve Square Roots
Practice 9.5.11. 5 x = 10
( 5 x ) 2 = ( 10) 2
5x = 100x = 20Check: 5 • 20 = 10
100 = 1010=10
2. 6 = 3 4 x ( 6 ) 2 = ( 3 4 x ) 2
36 = 9(4x)36 = 36xx = 1Check: 6 = 3 4 ( 1 ) 6 = 3 4 6 = 3•26 = 6
3. x + 5 = 3 ( x + 5 ) 2 = ( 3 ) 2
269CHAPTER 9 RADICAL EXPRESSIONS AND EQUATIONS
x + 5 =9x = 4Check: 4 + 5 = 3
9 = 3 3=3
4. 5 = x - 10( 5 ) 2 = ( x - 10 ) 2
25 = x - 10x = 35Check: x - 10 = 35 - 10 = 25 = 5
Practice 9.5.21. 5 x − 15 = - 5
5 x = 10( 5 x ) 2 = ( 10) 2
5x = 100x = 20Check: 5 • 20 - 15 = 100 − 15
2. 11 - 6 x = - 1 − 6 x = - 12( − 6 x ) 2 = ( - 12) 2
6x = 144x = 24Check: 11 -
6 ( 24) = 11- 144 = 11- 12= - 1
3.6 - x 5
+ 1 = 2
6 - x 5
= 1
5 • 6 - x 5
= 1 • 5
6 - x = 5 ( 6 - x ) 2 = ( 5 ) 2
6 -x = 25-x = 19x = -19
Check: 6 - ( - 19)
5 + 1 =
6 + 195
+ 1 =
255
+ 1 = 5 5
+ 1 = 1 + 1 = 2
4. x + 6 = 25x + 6 3
( x + 6 ) 2 = ( 25x + 6 3
) 2
x + 6 = 25x + 6 9
9(x+6) = 25x + 69x + 54 = 25x + 648 = 16xx = 3
Check: 3 + 6 = 25• 3 + 6 3
9 = 813
3 = 9/33 = 3
5. 5 x - 7 = 3 x + 1 ( 5 x - 7 ) 2 = ( 3 x + 1 ) 25(x-7) = 3x + 125x - 175 = 3x + 122x = 176x = 85 8 - 7 = 3 • 8 + 1 5 1 = 255 = 5
Practice 9.5.31. 30 - x = x
30 - x = x2 Square both sidesx2 + x - 30 = 0(x + 6)(x -5) = 0x + 6 = 0 or x - 5 = 0x = -6 or x = 5Check x = -6:
30 - ( - 6 ) = 36 = 6 6 is not a solution.Check x = 5: 30 - 5 = 25 = 5 5 is a solution.
2. 2 x + 4 = - 6 2x+4 = 36 Square both sides.2x = 32x = 16Check: 2 ( 16) + 4 = 36 = 6 ≠ - 6 16 is not a solution. There is nosolution.
3. 4 = 4 x + 7 - 3 = 4 x 9 = 4x Square both sides.x = 2.25Check: 4 ( 2 . 25) + 7 = 9 + 7 =
270 CHAPTER 9 RADICAL EXPRESSIONS AND EQUATIONS
3 + 7 = 10≠ 42.25 is not a solution. There is nosolution.
4. 2 x = 4 x + 8 4x2 =4x+8 Square both sides.4x2 - 4x - 8=04(x2 -x -2) = 04(x -2)(x+1) = 0x - 2 = 0 or x + 1 = 0x = 2 or x = -1Check x=2:2(2) = 4 ( 2 ) + 8 4 = 16 = 42 is a solution.Check x = -1:2(-1) = 4 ( - 1 ) + 8 -2 = 4 -2 ≠2-1 is not a solution.
Practice 9.5.41. d = 1 . 5 h
h = 6 + 12 = 18 feetd = 1 . 5 ( 18) d = 27d = 5.2 miles
2. d = 1 . 5 h 2.2 = 1 . 5 h 4.84= 1.5h Square both sides.h = 3.3 feet
3. d = 1 . 5 h 8 = 1 . 5 h 64 = 1.5h Square both sides.h = 42.7 feet42.7 - 7 = 35.7 feet for the platform.
Practice 9.5.51. v = µgR
v = 0 . 10( 9 . 8 ) 200v = 196v = 14 meters per second
2. v = µgR12 = 9 . 8 • 110µ 144 = 1078µµ = 0.13
Exercise Set 9.51. 2 x = 10
2x = 100 Square both sides.x = 50Check is left to student.
3. 4 = 2 w 16 = 2w Square both sides.w = 8
5. 5 2 x = 1525•2x = 225 Square both sides.50x = 225x = 4.5
7. 4 3 t = 2416•3t = 57648t = 576t = 12
9. 2 x + 3 = 5 2x + 3 = 25 Square both sides.2x = 22x = 11
11. 3 = 2 x - 5 9 = 2x - 5 Square both sides.2x = 14x = 7
13. y + 4 = 7 y + 4 = 49y = 45
15. x 2 − 5 = x x2 - 5 = x2
-5 = 0This is a contradiction. There is nosolution.
17. 7 x − 2 = 5 7 x = 7
7x = 49 Square both sides.x = 7
19. 8 = 3 x + 2 6 = 3 x 36 = 3x Square both sides.x = 12
21. 6 x − 7 = - 1
271CHAPTER 9 RADICAL EXPRESSIONS AND EQUATIONS
6 x = 6 6x = 36 Square both sides.x = 6
23. 6 x − 5 = 7 6 x = 12
6x = 144x = 24
25. 2 x − 10 = - 4 2 x = 6
2x = 36 Square both sidesx = 18
27. 7 - 2 x = 1 - 2 x = − 6 2x = 36 Square both sides.x = 18
29. 3 = - 2 + 5 x 5 = 5 x 25 = 5xx = 5
31. 2 = 4 + 6 x 2
− 3
5 = 4 + 6 x 2
10 = 4 + 6 x 100 = 4 + 6x Square both sides.96 = 6xx = 16
33.4 - 8 x 3
− 3 = - 1
4 - 8 x 3
= 2
4 - 8 x = 6 4 - 8x = 36 Square both sides.-8x = 32x = -4
35.6 - 5 x 2
− 5 = - 2
6 - 5 x 2
= 3
6 - 5 x = 6 6 - 5x = 36 Square both sides.-5x = 30
x = -6
37. 5 3 x - 20 = 11x + 1225(3x -20) = 11x + 12 Square bothsides.75x - 500 = 11x + 1264x = 512x = 8
39. 2 10x - 4 = 35x + 4 4(10x-4) = 35x + 440x - 16 = 35x + 45x = 20x = 4
41. y + 72 = y y + 72 = y2 Square both sides. y2 - y - 72 = 0(y -9)(y + 8) = 0y - 9 = 0 or y + 8 = 0y = 9 or y = -8Check: y = 9
9 + 72 = 81 = 9 = y 9 is a solution.Check y = -8:
− 8 + 72 = 64 = 8 ≠ - 8 -8 is not a solution.
43. 5 - 2 x = 7 - 2 x = 2 2x = 4 Square both sides.x = 2Check : 5 - 2 • 2 = 5 - 4 = 5 - 2 = 3 ≠ 7 2 is not a solution.There is nosolution.
45. x = 20 - x x2 = 20 -x Square both sides.x2 + x - 20 = 0(x + 5)(x - 4) = 0x + 5 = 0 or x - 4 = 0x = -5 or x = 4Check x = -5:-5 = 20 - ( - 5 ) -5 = 25-5 ≠5 -5 is not a solution.Check x = 4:4 = 20 - 4 4 = 16 4 = 4 4 is a solution.
272 CHAPTER 9 RADICAL EXPRESSIONS AND EQUATIONS
47. 4 - 16x = - 104-16x = 100 Square both sides.-16x = 96x = 6Check:
4 - 16( − 6 ) = 100 = 10≠ - 10There is no solution.
49. 2 x = 20x + 244x2 = 20x + 24 Square both sides.4x2 - 20x - 24 = 04(x2 - 5x - 6) = 04(x - 6)(x + 1) = 0x - 6 = 0 or x + 1 = 0x = 6 or x = -1Check x = 6:2(6) = 20• 6 + 2412 = 14412 = 126 is a solution.Check x = -1:2(-1) = 20( - 1 ) + 24-2 = 4 -2≠2-1 is not a solution.
51. p = 3 p + 18p2 = 3p +18 Square both sides.p2 -3p -18 = 0(p- 6)(p+3) = 0p - 6 = 0 or p + 3 = 0p = 6 or p = -3Check p = 6: 6 = 3 ( 6 ) + 186 = 366 = 6; 6 is a solution.Check p = -3: -3 = 3 ( - 3 ) + 18-3 = 9 -3 ≠ 3; -3 is not a solution.
53. − 4 = 6 x + 2 − 6 = 6 x 36 = 6x Square both sides.x = 6Check:
6 • 6 + 2 = 36 + 2 = 6 + 2 = 8 ≠ - 4 6 is not a solution.
55. h = 6 + 16 = 22 feetd = 1 . 5 ( 22) = 33 = 5 . 7 miles
57. 15 = 1 . 5 h 225 = 1.5h Square both sides.150 feet = h150 - 5 = 145 feet taller
59. d = 1 . 5 • 5 = 7 . 5 = 2 . 7 milesd = 1 . 5 • 4 = 6 . 0 = 2 . 4 miles2.7 - 2.4 = 0.3 miles farther
61.
56 feet
ScottJoline
106 feet
x
(56)2 +x2 =(106)2
3136 +x2 =11236
x2 -8100 = 0
(x + 90)(x - 90) = 0x + 90 = 0 or x - 90 = 0x = -90 or x = 90-90 is not valid so x = 90 feet is theheight of the kite.
62. m = 1 - 0 1 - 1
= 1 0
undefined.
63. y = mx + by = 1x - 2
64. m=1 - 1
6 - ( - 8 ) = 0
14= 0
Since m= 0 , the equation is y = 1
65. 4x2 - 27x + 18 = (4x -3)(x - 6)
66. 5x2 - 9x - 18 = (5x + 6)(x - 3)
273CHAPTER 9 RADICAL EXPRESSIONS AND EQUATIONS
67. 36x3y2z = 22•32x3y2z24xy4z2 = 23•3xy4z2
40x2y3z6 = 23•5x2y3z6
GCF = 22xy2z = 4xy2z
68. 8x2 - 8x = 08x(x-1) = 08x = 0 or x - 1 = 0x = 0 or x = 1
69. Eq 1: 3x + y = 2Eq 2: 6x + 2y = -5-2 times Eq 1: -6x -2y = -4Add together 0 = -9This is a contradiction. There is nosolution.
70. 2 + 1 b + 1
= b + 2 b + 1
2 ( b + 1 ) + ( b + 1 ) 1
b + 1 = b + 2
b + 1 ( b + 1 )
2b + 2 +1 = b + 22b + 3 = b + 2b = -1
Check: 2 + 1 - 1 + 1
= - 1 + 2 - 1 + 1
2 + 1/0 = 1/0 This is undefined. There is no solution.
71. x2 - 2x = x(x - 2)x2 - 4 = (x +2)(x -2)LCD = x(x - 2)(x + 2)
3
x 2 − 2 x + 1
x 2 − 4 =
3 x ( x - 2 )
+ 1 ( x + 2 ) ( x - 2 )
=
3 x ( x - 2 )
• x + 2 x + 2
+ x x
• 1
( x + 2 ) ( x - 2 ) =
3 ( x + 2 ) + x x ( x + 2 ) ( x - 2 )
= 3 x + 6 + x x ( x + 2 ) ( x - 2 )
=
4 x + 6 x ( x + 2 ) ( x - 2 )
72.( 2 xy3 ) 4 ( x 2 y ) 2
( 4 xy2 ) 2 = 16x 4 y 12x 4 y 2
16x 2 y 4 = x 6 y 10
73. 8 - 3(x-2) - (5 - 5x) = 8 - 3x + 6 - 5 + 5x = 9 + 2x
74. 5 - 9 + 2 ( 3 - 25 ) =5 - 3 + 2(3 - 5) = 2 + 2(-2) = 2 - 4 = -2
75.18s 4 s 5
6 s 7 = 9 • 2 s 4 s 5
6 s 7 =
9 • 4 • 2 s 6
6 s 7 = 3 • 2 s 3 2
6 s 7 = 2
s 4
76. 24 + 5 54 − 3 216 = 4 • 6 + 5 9 • 6 − 3 36• 6 =
2 6 + 5 • 3 6 − 3 • 6 6 = 2 6 + 15 6 − 18 6 = - 6
77. The x intercept is (-8,0) and the y-intercept is (0,2).
78. Yes, it passes the vertical line test.
274 CHAPTER 9 RADICAL EXPRESSIONS AND EQUATIONS
Section 9.6 Quadratic Equations
Practice 9.6.11. 3x2 + 7x - 6 = 0
a = 3, b = 7, c = -6
x = − 7 ± 7 2 − 4 ( 3 ) ( - 6 )
2 ( 3 )
x = − 7 ± 49 + 72
6
x = − 7 ± 121
6
x = − 7 ± 11
6 Solution 1 Solution 2x =(-7+11)/6 x = (-7 -11)/6x = 4/6 x = -18/6x = 2/3 x = -3
2. x2 - 5x + 1 = 0a = 1, b = -5, c = 1
x = − ( - 5 ) ± ( - 5 ) 2 − 4 ( 1 ) ( 1 )
2 ( 1 )
x =5 ± 25 − 4
2 = 5 ± 21
2 Solution 1 Solution 2
x = 5 + 21
2 x =
5 - 212
3. -x2 + 3x + 7 = 0a = -1, b = 3, c= 7
x = − 3 ± ( 3 ) 2 − 4 ( - 1 ) ( 7 )
2 ( - 1 )
x= − 3 ± 9 + 28
− 2 = − 3 ± 37
− 2 Solution 1 Solution 2
x = − 3 + 37
− 2 x =
− 3 - 37− 2
x = 3 - 37
2 x =
3 + 372
4. 2.1x2 - x - 5.4 =0a = 2.1, b = -1, c = -5.4
x = − ( - 1 ) ± ( - 1 ) 2 − 4 ( 2 . 1 ) ( - 5 . 4 )
2 ( 2 . 1 )
x = 1 ± 1 + 45. 36
4 . 2 = 1 ± 46. 36
4 . 2
Solution 1 Solution 2
x = 1 + 46. 36
4 . 2 x =
1 - 46. 364 . 2
x =1 + 6 . 80882
4 . 2 x=
1 - 6 . 808824 . 2
x =1.85924 x = -1.38305
Practice 9.6.21. 4x = 2x2 -1
2x2 - 4x - 1 = 0a = 2, b = -4, c = -1
x =− ( - 4 ) ± ( - 4 ) 2 − 4 ( 2 ) ( - 1 )
2 • 2
x=4 ± 16 + 8
4 = 4 ± 24
4 = 4 ± 2 6
4
x =4 4
± 2 6
4 = 1 ±
6 2
Solution 1 Solution 2
x = 1 + 6 2
x =1 - 6 2
2.1 4
= 1 2
x 2 − 3 x
0 = 1 2
x 2 − 3 x - 1 4
a = 1/2, b = -3 , c = -1/4
x=− ( - 3 ) ± ( - 3 ) 2 − 4 ( 1
2 ) ( - 1
4 )
2 ( 1
2 )
x=3 ± 9 + 1
2
1 = 3 ±
192
• 2 2
=
x = 3 ± 382
Solution 1 Solution 2
x =3 + 382
x=3 - 382
3. -2 = 4x2 - x4x2 - x + 2 = 0a = 4, b = -1, c = 2
x = − ( - 1 ) ± ( - 1 ) 2 − 4 ( 4 ) ( 2 )
2 ( 4 )
274 CHAPTER 9 RADICAL EXPRESSIONS AND EQUATIONS
x= 1 ± 1 − 32
8 = 1 ± − 31
8 There is no real solution. Theradicand is negative.
4. 20+ 2x =0.1x2 0.1x2 - 2x - 20 = 0a = 0.1, b = -2, c = -20
x =− ( - 2 ) ± ( - 2 ) 2 − 4 ( 0 . 1 ) ( - 20)
2 ( 0 . 1 )
x= 2 ± 4 + 8
0 . 2 = 2 ± 12
0 . 2 =
2 ± 2 3 0 . 2
= 10± 10 3
Solution 1 Solution 2 x= 10 + 10 3 x = 10 - 10 3
5.1 9
x 2 − 4 3
x = - 4
1 9
x 2 − 4 3
x + 4 = 0
a = 1/9, b = -4/3, c = 4
x =− ( − 4
3 ) ± ( − 4
3 ) 2 − 4 ( 1
9 ) ( 4 )
2 ( 1
9 )
x=
4
3 ± 16
9 − 16
9
2
9
= 4
3 ± 0
2
9
x = 4 3
• 9 2
= 6
Practice 9.6.41. a)y = -16x2 +60x + 7
y = -16(4)2 + 60(4) + 7y = -256 +240 +7 = -9 or 9 feet below ground levelb)40 = -16x2 + 60x + 7-16x2 + 60x - 33 = 0a = -16, b = 60, c=-33
x =− 60± ( 60) 2 − 4 ( - 16) ( - 33)
2 ( - 16)
x = − 60± 3600 − 2112
− 32=
x= − 60± 1488
− 32= − 60± 38. 57
− 32Solution 1 Solution 2
x =− 60 + 38. 57
− 32 x=
− 60 - 38. 57− 32
x=0.67 seconds x =3.08
secondsc) When the ball hits the ground theheight is 0.0 = -16x2 + 60x + 7a = -16, b = 60, c = 7
x=− 60± ( 60) 2 − 4 ( - 16) ( 7 )
2 ( - 16)
x=− 60± 3600 + 448
− 32=
x=− 60± 63. 62
− 32Solution 1 Solution 2
x=− 60 + 63. 62
− 32 x=
− 60 - 63. 62− 32
x=-0.11 (Invalid) x = 3.86 sec.
Exercise Set 9.61. 3x2 - 2x - 5 = 0
a = 3, b = -2, c = -5
x= − ( - 2 ) ± ( - 2 ) 2 − 4 ( 3 ) ( - 5 )
2 ( 3 )
x=2 ± 4 + 60
6 = 2 ± 64
6 = 2 ± 8
6 Solution 1 Solution 2x = -6/6 = -1 x = 10/6 =5/3
3. 0 = 2x2 + 5x + 2a = 2, b = 5, c = 2
x =− 5 ± 5 2 − 4 • 2 • 2
2 • 2
x=− 5 ± 25 − 16
4 = − 5 ± 9
4
x =− 5 ± 3
4 Solution 1 Solution 2x = -8/4 = -2 x = -2/4 = -1/2
5. x2 - 9x + 4 = 0a = 1, b = -9, c = 4
x =− ( - 9 ) ± ( - 9 ) 2 − 4 ( 1 ) ( 4 )
2 ( 1 )
x = 9 ± 81 − 16
2 = 9 ± 65
2 Solution 1 Solution 2
x = 9 + 65
2 x=
9 - 652
275CHAPTER 9 RADICAL EXPRESSIONS AND EQUATIONS
7. -4x2 +x+2=0a = -4, b = 1, c = 2
x =− 1 ± 1 2 − 4 ( - 4 ) ( 2 )
2 ( - 4 )
x =− 1 ± 1 + 32
− 8 = − 1 ± 33
− 8
Solution 1 Solution 2
x =− 1 + 33
− 8 x =
− 1 − 33− 8
x = 1 - 33
8 x =
1 + 338
9. 9x2 = 12x -49x2 - 12x + 4 =0a = 9, b = -12 , c = 4
x =− ( - 12) ± ( - 12) 2 − 4 ( 9 ) ( 4 )
2 ( 9 )
x = 12± 144 − 144
18= 12± 0
18
x = = 12
18=
2
3
11. 3x2 + 3x - 214
= 0
a = 3, b = 3, c = -21/4
x =− 3 ± 3 2 − 4 ( 3 ) ( − 21
4 )
2 ( 3 )
x=− 3 ± 9 + 63
6
x = − 3 ± 726
= − 3 ± 6 2 6
x= - 1 2
± 2
Solution 1 Solution 2
x = - 1 2
+ 2 x= - 1 2
- 2
13. -2.07x +0.05x2 -1.15 =00.05x2 - 2.07x - 1.15= 0a = 0.05, b = -2.07, c = -1.15
x = − ( - 2 . 07) ± ( - 2 . 07) 2 − 4 ( 0 . 05) ( - 1 . 15)
2 ( 0 . 05)
x= 2 . 07± 4 . 2849+ 0 . 23
0 . 1
x = 2 . 07± 4 . 51490 . 1
= 2 . 07± 2 . 12480 . 1
Solution 1 Solution 2
x = 2 . 07- 2 . 1248
0 . 1 x= 2 . 07+ 2 . 1248
0 . 1
x = - 0.548 x = 41.948
15. 4x2 + 2x = -24x2 +2x +2 =0a = 4, b = 2, c = 2
x=− 2 ± ( 2 ) 2 − 4 ( 4 ) ( 2 )
2 ( 4 )
x =− 2 ± 4 − 32
8 = − 2 ± − 24
8
The radicand is negative. There is no real solution.
17. -x +0.5x2 +0.5 =00.5x2 - x + 0.5 = 0a = 0.5, b = -1, c = 0.5
x =− ( - 1 ) ± ( - 1 ) 2 − 4 ( 0 . 5 ) ( 0 . 5 )
2 ( 0 . 5 )
x =1 ± 1 − 1
1 = 1 ± 0
1 = 1
1 = 1
19. a)y = 0.35x2 +5.4x +3877.85,000 = 0.35x2 +5.4x +3877.80=0.35x2 + 5.4x - 1122.2a = 0.35 , b = 5.4, c = -1122.2
x=− 5 . 4 ± ( 5 . 4 ) 2 − 4 ( 0 . 35) ( - 1122. 2 )
2 ( 0 . 35)
x = − 5 . 4 ± 29. 16+ 1571. 08
0 . 7
x =− 5 . 4 ± 1600. 24
0 . 7 = − 5 . 4 ± 40
0 . 7 x = 49.4 years; 1985 + 50 = 2035x = -64.9 years; 1985 - 65 = 1920 We will disregard this value since
our model is not valid for years prior to1985.b)2075 - 1985 = 90y = 0.35(90)2 + 5.4•90 + 3877.8y = 0.35(8100) +486 + 3877.8y = 2835 +486+3877.8 y = 7198.8 thousand birthsc)2000 - 1985 = 15y = 0.35(15)2 + 5.4(15) + 3877.8y = 0.35•225 +81 +3877.8y = 78.75 + 81+3877.8y = 4037.55 thousand births in 20004037.55 + 500 = 4537.55
276 CHAPTER 9 RADICAL EXPRESSIONS AND EQUATIONS
4537.55 = 0.35x 2 +5.4x+3877.80= 0.35x 2+ 5.4x - 659.75a = 0.35, b = 5.4, c =-659.75
x = − 5 . 4 ± ( 5 . 4 ) 2 − 4 ( 0 . 35) ( - 659. 75)
2 ( 0 . 35)
x =− 5 . 4 ± 29. 16 + 923. 65
0 . 7
x =− 5 . 4 ± 952. 81
0 . 7 = − 5 . 4 ± 30. 87
0 . 7 x = 36.4 years; 1985 + 37 = 2022Again we disregard the negativevalues for x, since our model is notvalid for years before 1985.
21. amount = rate • timeamount in = amount outLet x = time to fill75x = 60(x +1.5)75x = 60x + 9015x = 90x = 6 hours75 • 6 = 450 gallons
22.<---|----|----|----|----|----|----|---•----|----|--->
-2 -1.8 -1.6 -1.4 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2
23.
-9-8-7-6-5-4-3-2-10123456789
10
y
-10-9-8-7-6-5-4-3-2-10 1 2 3 4 5 6 7 8910x
Data #11
24.
n4 − 1
2 n - 7 2 n 5 − 7 n 4 − 2 n + 7 2 n 5 − 7 n 4 Subtract
0 - 2 n + 7 − 2 n + 7 Subtract
0
25. 25x2 -4 = (5x + 2)(5x + 2)
26. (s6t2) -2(s -6t-2) =s-12t-4s -6t-2 = s-18t-
6=1
s 18t 6
27.1 . 08 x10− 5
( 0 . 016) ( 0 . 00003) = 1 . 08 x10− 5
1 . 6 x 10− 2 ( 3 x 10− 5 ) 1 . 08 x10− 5
4 . 8 x 10− 7 = 0 . 225 x 102 =
2 . 25 x10− 1 x 102 = 2 . 25 x10
28.5 y 2
x 2 z •
3 x 5
10y 5 •
4 z 2
9 =
5 y 2
x 2 z •
3 x 5
2 • 5 y 5 •
2 • 2 z 2
3 • 3 = 2 x 3 z
3 y 3
29.2 x 2 + 2 x
x - 2 •
x 2 − 4
3 x 2 + 3 x =
2 x ( x + 1 ) x - 2
• ( x + 2 ) ( x - 2 )
3 x ( x + 1 ) = 2 ( x + 2 )
3
30.x 4 − x 3 − 2 x 2
x 2 + x ÷ ( x - 2 ) =
x 2 ( x 2 − x − 2 ) x ( x + 1 )
• 1
x - 2 =
x 2 ( x - 2 ) ( x + 1 ) x ( x + 1 )
• 1
x - 2 = x
31.1
x - 1 − 5 = 1
x - 1 − 5 •
x - 1 x - 1
=
1 - 5 ( x - 1 ) x - 1
= 1 - 5 x + 5 x - 1
= 6 - 5 x x - 1
32.3
t + 1 1
t
= t ( 3
t + 1 )
t ( 1
t )
= 3 + t 1
= 3 + t
33.4 ab2
9 = 2 b a
3
34. 2 • 3 • 5 2 = 5 2 • 3 = 5 6
35. 18x 7 y 2 = 2 • 3 2 x 7 y 2 = 3 x 3 y 2 x
277CHAPTER 9 RADICAL EXPRESSIONS AND EQUATIONS
36.4
6 = 4
6 •
6
6 = 4 6
6 = 2 6
3
37. -80 - 64 ÷22 •2(1-2) = -1-64 ÷4 •2(-1) = -1-16•(-2) =-1+32 =31
38. 3x2 + 5x - 2 = 0(3x - 1)(x + 2) = 03x - 1 = 0 or x + 2 = 03x = 1 or x = -2x = 1/3 or x = -2
39.z - 6 2
+ 1 = 2
z - 6 2
= 1
z - 6 = 2 z - 6 = 4 Square both sidesz = 10Check is left to student.
40. Eq 1: 3x - 4y = 2Eq 2: -6x+ 8y =52 times Eq 1: 6x - 8y = 4Add together 0 = 9This is a contradiction. There is nosolution.
41. Yes, it passed the vertical line test.
278 CHAPTER 9 RADICAL EXPRESSIONS AND EQUATIONS
Chapter 9 Review Exercises
1. − − 9 The radicand is negative.Not a real number.
2. − 16 = - 4
3. ( 49 − 36 ) 2 = ( 7 - 6 ) 2 = ( 1 ) 2 = 1
4.12 4 − 6 4
4 4 = 6 4
4 4 = 3
2
5. 49x 2 y 2 = 7 xy
6.9 m 2 n
16= 3 m n
4
7.50m 6 m
2 3 = 300m 2
2 3 =
3 • 100m 2
2 3 = 10m 3
2 3 = 5 m
8. 15x 16x = 15• 16x 2 = 4 x 15
9. 72 = 2 • 36 = 6 2
10.30025
= 12 = 3 • 4 = 2 3
11. 1568 = 2 5 • 7 2 = 2 2 • 7 2 = 28 2
12. 1440 = 122 • 10 = 12 10
13. x 2 y 4 z 11 = xy2 z 5 z
14. 4 f 3 h 9 = 4 fh4 fh
15. 108s 4 t 2 = 2 2 • 3 3 s 4 t 2 = 2 • 3 s 2 t 3 = 6 s 2 t 3
16. 700m 3 n 4 p 5 = 7 • 100m 3 n 4 p 5 = 10mn2 p 2 7 mp
17.3 t 294s 7
21s 4 = 3 t 2 • 3 • 7 2 s 7
21s 4 =
3 t•7 s 3 6 s
3 • 7 s 4 = t 6 s
s
18. 2 b 3 32b 5 cd8 = 2 b 3 2 5 b 5 cd8 = 2 b 3 • 2 b 2 d 4 2 bc = 8 b 5 d 4 2 bc
19.4 3 a 4 6 a 3 b 2
3 a 3 b = 4 18a 7 b 2
3 a 3 b =
4 9 • 2 a 7 b 2
3 a 3 b = 4 • 3 a 3 b 2 a
3 a 3 b = 4 2 a
20.5 x 3
2 y 5
20x
y 3 = 5 x 3 • 20x
2 y 5 y 3 =
50x 4
y 8 = 25• 2 x 4
y 8 = 5 x 2 2
y 4
21.4 6
12= 4 6
12•
3
3 = 4 18
36=
4 9 • 2 6
= 4 • 3 2 6
= 2 2
22.3 y
18y = 3 y
18y •
2 y
2 y =
279CHAPTER 9 RADICAL EXPRESSIONS AND EQUATIONS
3 y 2 y
36y 2 = 3 y 2 y
6 y = 2 y
2
23. ( 2 x + x ) 2 = ( 2 x ) 2 + 2 ( 2 x ) ( x ) + x 2 = 4 x + 4 x x + x 2
24. 6 t + 3 t − 2 t t = 9 t − 2 t t
25. 14 2 r − 2 14r + 3 14r − 11 2 r = 3 2 r + 14r
26. 12x 3 + 27x − x 75x = 4 • 3 x 3 + 3 3 x − x 25• 3 x =
2 x 3 x + 3 3 x − 5 x 3 x = 3 3 x − 3 x 3 x
27. 28 − 20 − 112 + 3 5 = 4 • 7 − 4 • 5 − 16• 7 + 3 5 =
2 7 − 2 5 − 4 7 + 3 5 = - 2 7 + 5
28. 3 t ( t − t ) + 3 ( t + t t ) =3 t - 3 t t + 3 t + 3 t t = 6 t
29. 2 y - 4 = 6 2y - 4 = 36 Square both sides.2y = 40y = 20
30. 4 16t = 164 • 4 t = 1616 t = 16
t = 1 t = 1 Square both sides.
31. − 4 t + 6 = 3 2 t + 8 -4t +6 =9(2t +8) Square both
sides.-4t + 6 = 18t + 72-22t = 66t = -3Check: − 4 ( - 3 ) + 6 = 3 2 ( - 3 ) + 8
18 = 3 2 9 • 2 = 3 2
3 2 = 3 2 -3 is a solution.
32. 3 - 4 - 3 x = - 2 - 4 - 3 x = - 5 4 - 3x = 25-3x = 21x = -73 - 4 - 3 ( - 7 ) = - 2 3 - 25 = - 2 3 - 5 = -2-2 = -2-7 is a solution.
33. 12 = 2 x + 14- 2 = 2 x 4 = 2xx = 212 = 2 ( 2 ) + 1412 = 2 + 1412 = 16This is a contadiction. There is nosolution.
34. x 2 + 5 + 5 = x
x 2 + 5 = x - 5 x2 + 5 = (x -5)2 Square both sides.x2 + 5 = x2 - 10x + 25-20 = -10xx = 2
2 2 + 5 + 5 = 2 9 + 5 = 2
3 + 5≠ 2This is a contradiction. There is nosolution.
35. -2x2 -3x +2 =0a = -2, b = -3, c = 2
x = − ( - 3 ) ± ( - 3 ) 2 − 4 ( - 2 ) ( 2 )
2 ( - 2 )
x= 3 ± 9 + 16
− 4 = 3 ± 25
− 4 = 3 ± 5
− 4 Solution 1 Solution 2
x = 3 + 5 − 4
x = 3 - 5 − 4
x = -2 x = 1/2
36. 4y2 -8y-3 =0a = 4, b = -8, c = -3
y = − ( - 8 ) ± ( - 8 ) 2 − 4 ( 4 ) ( - 3 )
2 ( 4 )
280 CHAPTER 9 RADICAL EXPRESSIONS AND EQUATIONS
y =8 ± 64 + 48
8 = 8 ± 112
8
y = 8 ± 4 7
8 = 1 ±
7 2
Solution 1 Solution 2y =
1 + 7 2
y =
1 - 7 2
37. q2 - 2 3
q + 1 9
= 0
a = 1, b = -2/3, c = 1/9
q = - − 2
3 ± ( − 2
3 ) 2 − 4 ( 1 ) ( 1
9 )
2 ( 1 )
q =
2
3 ± 4
9 − 4
9
2 =
2
3 ± 0
2
q = 2
3 •
1
2 =
1
3
38. b2 = b+1.75b2 - b - 1.75 = 0a = 1, b = -1, c = -1.75
b = − ( - 1 ) ± ( - 1 ) 2 − 4 ( 1 ) ( - 1 . 75)
2 ( 1 )
b = 1 ± 1 + 7
2 = 1 ± 8
2
b = 1 ± 2 2
2 = 1
2 ± 2
Solution 1 Solution 2
b = 1 2
+ 2 b = 1 2
− 2
39. distance = rate •timedistance/rate = time
200 feet
100 feetd
(100)2 + (200)2 = d2
10000 +40000= d2
d2 -50000 =0a = 1, b = 0, c = -50000
d = − 0 ± 0 2 − 4 ( 1 ) ( - 50000)
2 ( 1 )
d=0 ± 0 + 200000
2 = ± 447. 2
2 d = 223.6 The negative value isinvalid since we can’t have a
negative distance.100 + 200 - 223.6 = 76.4 more feet76.4 ÷4.4 = 17 extra seconds to
walk the edge.
281CHAPTER 9 RADICAL EXPRESSIONS AND EQUATIONS
Chapter 9 Test
1. 16 - 2 16 − ( 2 25 + 1 ) − 3
=
16 - 2 • 4 − ( 2 • 5 + 1 ) − 3
=
16 - 8 − ( 10 + 1 ) − 3
= 16 - 8 − 11− 3
=
− 3 − 3
= 1
2.36t
a 2 = 6 t
a
3.20y 4
5 y = 4 • 5 y 4
5 y =
2 • 2 5 y
5 y = 4
4. 6 5 t 3 + t 45t + 80t 3 = 6 t 5 t + t 9 • 5 t + t 16• 5 t = 6 t 5 t + 3 t 5 t + 4 t 5 t = 13t 5 t
5. 588 = 2 2 3 • 7 2 = 2 • 7 3 = 14 3
6. h 11j 7 = h 5 j 3 hj
7. 15 ( 30 − 6 ) - 3 ( 5 2 + 10 ) =15• 15• 2 − 15• 6 − 15 2 − 3 10 = 152 • 2 − 90 − 15 2 − 3 10 =
15 2 − 9 • 1 0 − 15 2 − 3 10 = − 3 10 − 3 10 = - 6 10
8. 9 n 2 + 12n + 4 = ( 3 n + 2 ) 2 = | 3 n + 2 |
9.15a 2 b
c 3
10b 3
c =
3 • 5 • 2 • 5 a 2 b 4
c 4 = 3 • 2 • 5 2 a 2 b 4
c 4 =
5 ab2 6
c 2
10. − 1 4
2 y + 1 3
2 y + 1 2
2 y =
− 1 4
• 3 3
2 y + 1 3
• 4 4
2 y + 1 2
• 6 6
2 y =
− 3 12
2 y + 4 12
2 y + 6 12
2 y =
7 12
2 y
282 CHAPTER 9 RADICAL EXPRESSIONS AND EQUATIONS
11. 6 y 42y = 6 y•6 • 7 y = 6 2 • 7 y 2 = 6 y 7
12.3 d
24= 3 d
4 • 6 •
6
6 = 3 d 6
4 • 6 2 =
3 d 6 2 • 6
= d 6 4
13. 3 2 x = 129(2x) = 144 Square both sides18x = 144x = 8Check left for student.
14. x 2 + 4 x + 4 − 1 = 2 x
x 2 + 4 x + 4 = 2 x + 1 x2 + 4x + 4 = (2x + 1)2 Square bothsides.
x2
+4x+4 =4x2
+4x+10 =
3x2 - 30 =
3(x + 1)(x -1)x + 1 = 0 or x - 1 = 0x = -1 or x = 1Check x = -1:
( - 1 ) 2 + 4 ( - 1 ) + 4 − 1 = 2 ( - 1 ) 1 - 4 + 4 − 1 = - 2
1 -1 ≠ -2; -1 is not a solution.Check x = 1
( 1 ) 2 + 4 ( 1 ) + 4 − 1 = 2 ( 1 ) 9 − 1 = 2
3 - 1 = 2 ; 1 is a solution.
15. 3 x + 1 = 2 x - 5 9(x+1) = 2x - 5 Square both
sides.9x + 9 = 2x - 57x = -14x = -2Check: 3 - 2 + 1 = 2 ( − 2 ) - 5 3 − 1 = − 9 The radicand is negative. There is
no real solution.
16. 6x2 +13x+6 =0a = 6, b = 13, c = 6
x = − 13± ( 13) 2 − 4 ( 6 ) ( 6 )
2 ( 6 )
x = − 13± 169 − 14412
= − 13± 2512
x = − 13± 5
12Solution 1 Solution 2
x = − 13 + 5
12 x =
− 13 - 5 12
x = -8/12 = -2/3 x = -18/12 = -3/2
17. 3y2 - 3y -1 = 0a = 3, b = -3, c = -1
y = − ( - 3 ) ± ( - 3 ) 2 − 4 ( 3 ) ( - 1 )
2 ( 3 )
y = 3 ± 9 + 12
6 = 3 ± 21
6
18.4 feet
7 feet
d
42 + 72 = d2
d2 = 16 + 49 d2 = 65d2 - 65 = 0a = 1, b = 0, c = -65
d = 0 ± 0 - 4 ( 1 ) ( - 65)
2 ( 1 )
d = ± 260
2 = ± 16. 12
2 d = 8.06 feet (Disregard the
negative value. It is invalid.)
283CHAPTER 9 RADICAL EXPRESSIONS AND EQUATIONS
284 CHAPTER 9 RADICAL EXPRESSIONS AND EQUATIONS