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361Algebra 2

Worked-Out Solution Key

Prerequisite Skills (p. 412)

1. The square roots of 81 are 9 and 29.

2. In the expression 25, the exponent is 5.

3. For the polynomial function whose graph is shown, the sign of the leading coeffi cient is positive.

4. 5x2y

} 15x3y21 5

5x2 2 3y1 2 (21)

}} 15

5 5x21y 2

} 15

5 y2

} 3x

5. 32x23y4

} 24x23y22 p

3x }

9y 5

96x23 1 1y 4

}} 216x23y22 1 1

5 4x22y4

} 9x23y21

5 4x22 2 (23)y4 2 (21)

}} 9

5 4xy5

} 9

6. (2x5y23)23 5 1 }

(2x5y23)3

5 1 }

23(x5)3(y23)3

5 1 }

8x15y29

5 y9

} 8x15

7. 22x 2 5y 5 10

25y 5 2x 1 10

y 5 2 2 } 5 x 2 2

8. x 2 1 } 3 y 5 21

3x 2 y 5 23

2y 5 23x 2 3

y 5 3x 1 3

9. 8x 2 4xy 5 3

24xy 5 28x 1 3

y 5 2 2 3 } 4x

10. f (x) 5 x3 2 4x 1 6

x 23 22 21 0 1 2 3

f (x) 29 6 9 6 3 6 21

The degree is odd and the leading coeffi cient is positive, so f (x) → 2` as x → 2` and f (x) → 1 ` as x → 1 `.

2

x

y

21

11. f (x) 5 2x5 1 7x2 1 2

x 22 21 0 1 2

f (x) 62 10 2 8 22

The degree is odd and the leading coeffi cient is negative, so f (x) → ` as x → 2` and f (x) → 2` as x → 1`.

2

x

y

21

12. f (x) 5 x4 2 4x2 1 x

x 23 22 21 0 1 2 3

f (x) 42 22 24 0 22 2 48

The degree is even and the leading coeffi cient is positive, so f (x) → 1` as x → 2` and f (x) → 1 ` as x → 1 `.

2

x

y

21

Lesson 6.1

6.1 Guided Practice (pp. 415–416)

1. Because n 5 4 is even and a 5 625 > 0, 625 has two real fourth roots. Because 54 5 625 and (25)4 5 625, you

can write 6 4 Ï}

625 5 65 or 66251/4 5 65.

2. Because n 5 6 is even and a 5 64 > 0, 64 has two real 6th roots. Because 26 5 64 and (22)6 5 64, you can

write 6 6 Ï}

64 5 62 or 6641/6 5 62.

3. Because n 5 3 is odd and a 5 264 < 0, 264 has one real cube root. Because (24)3 5 264, you can write

3 Ï}

264 5 24 or (264)1/3 5 24.

4. Because n 5 5 and a 5 243 > 0, 243 has one real fi fth

root. Because 35 5 243, you can write 5 Ï}

243 5 3 or

2431/5 5 3.

5. 45/2 5 (41/2)5 5 25 5 32 6. 921/2 5 1 }

91/2 5 1 }

Ï}

9 5

1 } 3

7. 813/4 5 ( 4 Ï}

81 )3 5 33 5 27 8. 17/8 5 (11/8)7 5 17 5 1

9. 42/5 ø 1.74 10. 6422/3 ø 0.06

11. ( 4 Ï}

16 )5 5 32 12. ( 3 Ï

}

230 )2 ø 9.65

13. x3 5 64 14. 1 }

2 x5 5 512

x 5 3 Ï}

64 x5 5 1024

x 5 4 x 5 5 Ï}

1024

x 5 4

Chapter 6

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362Algebra 2Worked-Out Solution Key

15. 3x2 5 108 16. 1 }

4 x3 5 2

x2 5 36 x3 5 8

x 5 6 Ï}

36 x 5 3 Ï}

8

x 5 66 x 5 2

17. (x 2 2)3 5 214 18. (x 1 5)4 5 16

x 2 2 5 3 Ï}

214 x 1 5 5 6 4 Ï}

16

x 5 3 Ï}

214 1 2 x 5 6 4 Ï}

16 2 5

x ø 20.41 x 5 23 or

x 5 27

19. w 5 0.0167l3

a. 275 5 0.0167l3

16,467 ø l3

3 Ï}

16,467 ø l 25.4 ø l A coral cod that weighs 275 grams is about

25 centimeters long.

b. 340 5 0.0167l3

20,359 ø l3

3 Ï}

20,359 ø l 27.3 ø l A coral cod that weighs 340 grams is about


c. 450 5 0.0167l3

26,946 ø l3

3 Ï}

26,946 ø l 30 ø l A coral cod that weighs 450 grams is about


6.1 Exercises (pp. 417–419)

Skill Practice

1. In the expression 4 Ï}

10,000 , the number 4 is called the index of the radical.

2. If a < 0, then there are no real fourth roots. If a > 0, then

there are two real fourth roots: 6 4 Ï}

a 5 6a1/4. If a < 0 or

if a > 0, then there is one real fi fth root: 5 Ï}

a 5 a1/5.

3. C; 21/3 5 3 Ï}

2 4. A; 23/2 5 ( Ï}

2 )3

5. D; 22/3 5 ( 3 Ï}

2 )2 6. B; 21/2 5 Ï

}

2

7. 3 Ï}

12 5 121/3 8. 5 Ï}

8 5 81/5

9. ( 3 Ï}

10 )7 5 107/3 10. ( 8 Ï

}

15 )3 5 153/8

11. 51/4 5 4 Ï}

5 12. 71/3 5 3 Ï}

7

13. 142/5 5 ( 5 Ï}

14 )2 14. 219/4 5 ( 4 Ï

}

21 )9

15. Because n 5 2 is even and a 5 64 > 0, 64 has two real square roots. Because 82 5 64 and (28)2 5 64, you can

write 6 Ï}

64 5 68 or 6641/2 5 68.

16. Because n 5 3 is odd and a 5 227 < 0, 227 has one real cube root. Because (23)3 5 227, you can

write 3 Ï}

227 5 23 or (227)1/3 5 23.

17. Because n 5 4 is even and a 5 0, 0 has one real fourth

root. Because 04 5 0, you can write 4 Î}

0 5 0.

18. Because n 5 3 is odd and a 5 343 > 0, 343 has one real

cube root. Because 73 5 343, you can write 3 Ï}

343 5 7 or 3431/3 5 7.

19. Because n 5 4 is even and a 5 216 < 0, 216 has no real fourth roots.

20. Because n 5 5 is odd and a 5 232 < 0, 232 has one real fi fth root. Because (22)5 5 232, you can

write 5 Ï}

232 5 22 or (232)1/5 5 22.

21. 6 Ï}

64 5 62 22. 81/3 5 2

23. 163/2 5 (161/2)3 24. 3 Ï}

2125 5 25

5 43 5 64

25. 272/3 5 (271/3)2 26. (2243)1/5 5 23

5 32 5 9

27. ( 3 Ï}

8 )22 5

1 }

( 3 Ï}

8 )2 5 1 }

22 5 1 } 4

28. ( 3 Ï}

264 )4 5 (24)4 5 256

29. ( 4 Ï}

16 )27 5

1 }

( 4 Ï}

16 )7 5 1 }

27 5 1 } 128

30. 253/2 5 (251/2)3 5 53 5 125

31. 6422/3 5 1 }

642/3 5 1 }

(641/3)2 5

1 }

42 5 1 } 16

32. 1 }

8123/4 5 813/4 5 (811/4)3 5 33 5 27

33. C; 1285/7 5 (1281/7)5 5 25 5 32

34. 5 Ï}

32,768 5 8 35. 7 Ï}

1695 ø 2.89

36. 9 Ï}

2230 ø 21.83 37. 851/6 ø 2.10

38. 2521/3 ø 0.34 39. 20,7361/4 5 12

40. ( 4 Ï}

187 )3 ø 50.57 41. ( Ï

} 6 )25 ø 0.01

42. ( 5 Ï}

28 )8 ø 27.86 43. 8625/6 ø 0.02

44. 19742/7 ø 8.74

45. 1 }

(217)3/5 5 (217)23/5 ø 20.18

46. B; 273/5 ø 7.22 47. Sample answer:

53/2 ø 11.18 7291/6; 271/3

3 Ï}

81 ø 4.33

( 3 Ï}

2 )8 ø 6.35

48. The cube root of 27 is 3, not 9.

x3 5 27

x 5 3 Ï}

27

x 5 3

49. The negative fourth root of 81 was not included.

x4 5 81

x 5 6 4 Ï}

81

x 5 63

Chapter 6, continued

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363Algebra 2


50. x3 5 125` 51. 5x3 5 1080

x 5 3 Ï}

125 x3 5 216

x 5 5 x 5 3 Ï}

216

x 5 6

52. x6 1 36 5 100

x6 5 64

x 5 6 6 Ï}

64

x 5 62

53. (x 2 5)4 5 256

x 2 5 5 6 4 Ï}

256

x 5 6 4 Ï}

256 1 5

x 5 4 1 5 or x 5 24 1 5

x 5 9 or x 5 1

54. x5 5 248 55. 7x4 5 56

x 5 5 Ï}

248 x4 5 8

x ø 22.17 x 5 6 4 Ï}

8

x ø 61.68

56. x3 1 40 5 25 57. (x 1 10)5 5 70

x3 5 215 x 1 10 5 5 Ï}

70

x 5 3 Ï}

215 x 5 5 Ï}

70 2 10

x ø 22.47 x ø 27.66

58. x6 2 34 5 181

x6 5 215

x 5 6 6 Ï}

215

x ø 62.45

59. a. When a < 0, you can see from the graph that the line y 5 a does not intersect the graph of y 5 xn. There are no real nth roots.

When a 5 0, you can see from the graph that the line y 5 a intersects the graph of y 5 xn once. There is one real nth root.

When a > 0, you can see from the graph that the line y 5 a intersects the graph of y 5 xn twice. There are two real nth roots.

b.

x

y

y 5 aa > 0

y 5 0y 5 aa < 0

Problem Solving

60. V 5 4 } 3 πr3

905 5 4 } 3 πr3

216.05 ø r3

3 Ï}

216.05 ø r

6 ø r The radius of the shot is about 6 centimeters.

61. S 5 4πr2

232 5 4πr2

18.46 ø r2

6 Ï}

18.46 ø r 64.3 ø r Reject the negative value, 24.3. The radius of the

bowling ball is about 4.3 inches.

62. r 5 1 p2

} p1 2 1/n

2 1

Butter: r 5 1 2.195 }

0.7420 2 1/40

2 1

r ø (2.96)1/40 2 1

r ø 1.027 2 1

r ø 0.027 5 2.7%

Chicken: r 5 1 1.087 } 0.4430 2 1/40

2 1

r ø (2.45)1/40 2 1

r ø 1.023 2 1

r ø 0.023 5 2.3%

Eggs: r 5 1 1.356 } 0.6710 2 1/40

2 1

r ø (2.02)1/40 2 1

r ø 1.018 2 1

r ø 0.018 5 1.8%

Sugar: r 5 1 0.4560 } 0.0936 2 1/40

2 1

r ø (4.87)1/40 2 1

r ø 1.040 2 1

r ø 0.040 5 4.0%

63. p 5 ks3

1.2 5 k(1700)3

1.2

} 17003 5 k

1 }}

4,094,166,667 5 k

When p 5 1.5: 1.5 5 1 }} 4,094,166,667 s3

6,141,250,000 5 s3

3 Ï}}

6,141,250,000 5 s

1831 ø s The speed of the fan if it uses 1.5 horsepower is about

1831 revolutions per minute.

64. Q 5 3.367lh3/2

Q 5 3.367(20)(5)3/2

Q ø 753

The fl ow rate of the weir is about 753 cubic feet per second.


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65. a. V 5 s3

V 5 (16)3

V 5 4096

The volume of the cube is 4096 cubic millimeters.

b. Tetrahedron: V 5 0.118x3

4096 5 0.118x3

34,712 ø x3

3 Ï}

34,712 ø x 33 ø x Edge length: about 33 millimeters

Octahedron: V 5 0.471x3

4096 5 0.471x3

8696 ø x3

3 Ï}

8696 ø x 21 ø x Edge length: about 21 millimeters

Dodecahedron: V 5 7.663x3

4096 5 7.663x3

535 ø x3

3 Ï}


Icosahedron: V 5 2.182x3

4096 5 2.182x3

1877 ø x3

3 Ï}


c. No; the icosahedron has the greatest number of faces but not the smallest edge length. Of the three polyhedra with equilateral triangles for faces, the icosahedron has the smallest edge length, but the dodecahedron’s faces are regular pentagons, so its edge length is smaller.

66. An equation for mass m in terms of speed s is m 5 ks6, where k is a constant.

At speed of 1 meter per second: m 5 k(1)6 5 k

Twice as massive: 2k 5 ks6

2 5 s6

6 6 Ï}

2 5 s

61.1 ø s Reject the negative value, 21.1. The river’s speed must

be about 1.1 meters per second to transport particles that are twice as massive as usual.

10 times as massive: 10k 5 ks6

10 5 s6

6 6 Ï}

10 5 s

61.5 ø s

Reject the negative value, 21.5. The river’s speed must be about 1.5 meters per second to transport particles that are 10 times as massive as usual.

100 times as massive: 100k 5 ks6

100 5 s6

6 6 Ï}

100 5 s

62.2 ø s Reject the negative value 22.2. The river’s speed must be

about 2.2 meters per second to transport particles that are 100 times as massive as usual.

Mixed Review

67. x 5 3, y 5 5 68. x 5 6, y 5 22

x 1 3y

} x 2 y

5 3 1 3(5)

} 3 2 5 4x 2 y

} x 2 2y

5 4(6) 2 (22)

} 6 2 2(22)

5 18

} 22 5 29 5

26 } 10 5

13 } 5

69. f (x) 5 x2 2 2x 2 35 5 (x 2 7)(x 1 5)

The zeros are 7 and 25.

70. f (x) 5 x2 2 8x 1 25

x 5 2(28) 6 Ï

}}

(28)2 2 4(1)(25) }}}

2(1)

x 5 8 6 Ï

}

236 } 2

x 5 8 6 6i

} 2

x 5 4 6 3i

The zeros are 4 1 3i and 4 2 3i.

71. f (x) 5 x3 2 8x2 1 4x 2 32

x 5 61, 62, 64, 68, 616, 632

Test x 5 8:

8 1 28 4 232

8 0 32

1 0 4 0

f (x) 5 (x 2 8)(x2 1 4) x2 1 4 5 0

x2 5 24

x 5 6 Ï}

24

x 5 62i

f (x) 5 (x 2 8)(x 2 2i)(x 1 2i)

The zeros are 8, 2i, and 22i.

72. f (x) 5 x3 2 4x2 1 25x 1 100

x 5 61, 62, 64, 65, 610, 620, 625, 650, 6100

Test x 5 24:

24 1 4 25 100

24 0 2100

1 0 25 0

f (x) 5 (x 1 4)(x2 1 25) 5 (x 1 4)(x 2 5i)(x 1 5i)

The zeros are 24, 5i, and 25i.


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365Algebra 2


73. f (x) 5 x4 2 3x3 2 31x2 1 63x 1 90

x 5 61, 62, 63, 65, 66, 69, 610, 615, 618, 630, 645, 690

Test x 5 3:

3 1 23 231 63 90

3 0 293 290

1 0 231 230 0

f (x) 5 (x 2 3)(x3 2 31x 2 30) x 5 61, 62, 63, 65, 66, 610, 615, 630

Test x 5 25:

25 1 0 231 230

25 25 30

1 25 26 0

f (x) 5 (x 2 3)(x 1 5)(x2 2 5x 2 6) 5 (x 2 3)(x 1 5)(x 2 6)(x 1 1)

The zeros are 3, 25, 6, and 21.

74. f (x) 5 x4 1 10x3 1 25x2 2 36

x 5 61, 62, 63, 64, 66, 69, 612, 618, 636

Test x 5 1:

1 1 10 25 0 236

1 11 36 36

1 11 36 36 0

f (x) 5 (x 2 1)(x3 1 11x2 1 36x 1 36) Test x 5 22:

22 1 11 36 36

22 218 236

1 9 18 0

f (x) 5 (x 2 1)(x 1 2)(x2 1 9x 1 18) 5 (x 2 1)(x 1 2)(x 1 6)(x 1 3)

The zeros are 1, 22, 26, and 23.

75. x24

} x3 5 x24 2 3 Quotient of powers property

5 x27

5 1 }

x7 Negative exponent property

76. (x4)23 5 x212 Power of a power property

5 1 }

x12 Negative exponent property

77. (3x2y)23 5 1 }

(3x2y)3 Negative exponent property

5 1 }

33x6y3 Power of a product property

5 1 }

27x6y3

78. 4x0y24 5 4y24 Zero exponent property

5 4 }

y4 Negative exponent property

79. x6 p x22 5 x6 1 (22) Product of powers property

5 x4

80. 1 x3 }

y22 2 2 5

(x3)2

} ( y22)2

Power of a quotient property

5 x6

} y24 Power of a power property

5 x6y4 Negative exponent property

81. 4x3y6

} 10x5y23 5

4x3 2 5y6 2 (23)

}} 10 Quotient of powers property

5 2x22y9

} 5

5 2y9

} 5x2 Negative exponent property

82. 3x

} x3y2 p

y4

} 9x22 5

3xy4

} 9x3 1 (22)y2 Product of powers property

5 3xy4

} 9xy2

5 3x1 2 1y4

} 9y2 Quotient of powers property

5 3y4

} 9y2 Zero exponent property

5 3y4 2 2

} 9 Quotient of powers property

5 y2

} 3

Lesson 6.2


1. (51/3 p 71/4)3 5 (51/3)3 p (71/4)3

5 5(1/3 p 3) p 7(1/4 p 3)

5 5 p 73/4

2. 23/4 p 21/2 5 2(3/4 1 1/2) 5 25/4

3. 3 }

31/4 5 31

} 31/4 5 3(1 2 1/4) 5 33/4

4. 1 201/2 }

51/2 2 3 5 F 1 20

} 5 2 1/2

G 3 5 (41/2)3 5 23 5 8

5. S 5 km2/3

5 8.4(9.5 3 104)2/3

5 8.4(9.5)2/3(104)2/3

ø 8.4(4.49)(108/3) ø 17,506

The sheep’s surface area is about 17,506 square centimeters.

6. 4 Ï}

27 p 4 Ï}

3 5 4 Ï}

27 p 3 5 4 Ï}

81 5 3

7. 3 Ï}

250 }

3 Ï}

2 5 3 Î}

250

} 2 5

3 Ï}

125 5 5

8. 5 Î}

3 }

4 5

5 Ï}

3 }

5 Ï}

4 p

5 Ï}

8 }

5 Ï}

8 5

5 Ï}

24 }

5 Ï}

32 5

5 Ï}

24 } 2


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9. 3 Ï}

5 1 3 Ï}

40 5 3 Ï}

5 1 3 Ï}

8 p 3 Ï}

5

5 3 Ï}

5 1 2 3 Ï}

5

5 (1 1 2) 3 Ï}

5 5 3 3 Ï}

5

10. 3 Ï}

27q9 5 3 Ï}

33(q3)3 5 3 Ï}

33 p 3 Ï}

(q3)3 5 3q3

11. 5 Î}

x10

} y5 5

5 Ï}

x10 }

5 Ï}

y5 5

5 Ï}

(x2)5 }

5 Ï}

y5 5

x2

} y

12. 6xy3/4

} 3x1/2y1/2 5 2x(1 2 1/2)y(3/4 2 1/2) 5 2x1/2y1/4

13. Ï}

9w5 2 w Ï}

w3 5 3w2 Ï}

w 2 w2 Ï}

w

5 (3w2 2 w2) Ï}

w 5 2w2 Ï}

w


Skill Practice

1. No, 2 Ï}

5 and 2 3 Ï}

5 are not like radicals because they do not have the same index. The expression 2 Ï

} 5 has an

index of 2 and 2 3 Ï}

5 has an index of 3.

2. A radical expression with index n is in simplest form if the radicand has no perfect nth powers as factors and any denominator has been rationalized.

3. 53/2 p 51/2 5 5(3/2 1 1/2) 5 52 5 25

4. (62/3)1/2 5 6(2/3 p 1/2) 5 61/3

5. 31/4 p 271/4 5 (3 p 27)1/4 5 811/4 5 3

6. 9 }

924/5 5 9(1 1 4/5) 5 99/5

7. 801/4

} 521/4 5 801/4 p 51/4

5 (80 p 5)1/4

5 4001/4

5 161/4 p 251/4

5 2 p 251/4

5 2 p 51/2

8. 1 73 }

43 2 21/3

5 F 1 7 } 4 2 3 G 21/3

5 1 7 } 4 2 [3 p (21/3)]

5 1 7 } 4 2 21

5 4 } 7

9. 112/5

} 114/5 5 11(2/5 2 4/5) 5 1122/5 5

1 }

112/5 p 113/5

} 113/5 5

5 Ï}

1331 } 11

10. (123/5 p 83/5)5 5 [(12 p 8)3/5]5

5 (963/5)5 5 963 5 884,736

11. 12022/5 p 1202/5

}} 723/4 5

120(22/5 1 2/5)

} 723/4 5

1 }

723/4 5 73/4

12. 645/9 p 642/9

} 43/4 5

(641/3)5/3 p (641/3)2/3

}} 43/4

5 47/3

} 43/4

5 47/3 2 3/4

5 419/12

5 4 p 47/12

5 4 p (47/2)1/6

5 4 p (27)1/6

5 4(2 p 26)1/6

5 8 p 21/6

13. (165/9 p 57/9)23 5 (165/9)23 p (57/9)23

5 1625/3 p 527/3

5 1 }

165/3 p 57/3 p 161/3 p 52/3

} 161/3 p 52/3

5 161/3 p (52)1/3

} 162 p 53

5 4001/3

} 32,000

5 81/3 p 501/3

} 32,000

5 3 Ï}

50 } 16,000

14. 133/7

} 135/7 5 13(3/7 2 5/7) 5 1322/7 5

1 }

132/7 p 135/7

} 135/7 5

7 Ï}

371,293 } 13

15. Ï}

20 p Ï}

5 5 Ï}

20 p 5 5 Ï}

100 5 10

16. 3 Ï}

16 p 3 Ï}

4 5 3 Ï}

16 p 4 5 3 Ï}

64 5 4

17. 4 Ï}

8 p 4 Ï}

8 5 4 Ï}

8 p 8 5 4 Ï}

64 5 4 Ï}

16 p 4

5 4 Ï}

16 p 4 Ï}

4 5 2 4 Ï}

4 5 2 Ï}

2

18. ( 3 Ï}

3 p 4 Ï}

3 )12 5 1 31/3 p 31/4 2 12

5 312/3 p 312/4

5 34 p 33 5 37 5 2187

19. 5 Ï}

64 }

5 Ï}

2 5 5 Î}

64

} 2 5

5 Ï}

32 5 2

20. Ï

}

3 }

Ï}

75 5 Î}

3 } 75 5 Î}

1 }

25 5

Ï}

1 }

Ï}

25 5

1 } 5

21. 4 Ï}

36 p 4 Ï}

9 }

4 Ï}

4 5

4 Ï}

36 p 9 }

4 Ï}

4 5

4 Ï}

324 }

4 Ï}

4 5 4 Î}

324

} 4 5

4 Ï}

81 5 3

22. 4 Ï}

8 p 4 Ï}

16 }

8 Ï}

2 p 8 Ï}

3 5

4 Ï}

128 }

8 Ï}

6 p 6

7/8 }

67/8

5 1281/4 p 67/8

} 6

5 (1282)1/8 p (67)1/8

}} 6

5 (1282 p 67)1/8

} 6

5 (4,586,471,424)1/8

}} 6

5 4(69,984)1/8

} 6

5 2 8 Ï}

69,984 } 3

23. C;

3 4 Ï}

32 p (26 4 Ï}

5 ) 5 3 4 Ï}

16 p 2 p (26 4 Ï}

5 ) 5 3

4 Ï}

16 p 4 Ï}

2 p (26 4 Ï}

5 ) 5 3 p 2 p

4 Ï}

2 p (26 4 Ï}

5 ) 5 236

4 Ï}

2 p 5

5 236 4 Ï}

10

24. Ï}

72 5 Ï}

36 p 2 5 Ï}

36 p Ï}

2 5 6 Ï}

2

25. 6 Ï}

256 5 6 Ï}

64 p 4 5 6 Ï}

64 p 6 Ï}

4 5 2 6 Ï}

4 5 2 3 Ï}

2


n2ws-06a.indd 366 6/27/06 10:35:30 AM

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367Algebra 2


26. 3 Ï}

108 p 3 Ï}

4 5 3 Ï}

108 p 4

5 3 Ï}

432

5 3 Ï}

216 p 2

5 3 Ï}

216 p 3 Ï}

2

5 6 3 Ï}

2

27. 5 4 Ï}

64 p 2 4 Ï}

8 5 5 p 2 4 Ï}

64 p 8

5 10 4 Ï}

512

5 10 4 Ï}

256 p 2

5 10 4 Ï}

256 p 4 Ï}

2

5 10 p 4 p 4 Ï}

2

5 40 4 Ï}

2

28. 3 Î}

1 }

6 5

3 Ï}

1 }

3 Ï}

6 p

3 Ï}

36 }

3 Ï}

36 5

3 Ï}

36 }

3 Ï}

216 5

3 Ï}

36 } 6

29. 3 }

4 Ï}

144 5

3 }

4 Ï}

16 p 9 30. 6 Î}

81

} 4 5

6 Ï}

81 }

6 Ï}

4 p

6 Ï}

16 }

6 Ï}

16

5 3 }

4 Ï}

16 p 4 Ï}

9 5

6 Ï}

1296 }

6 Ï}

64

5 3 }

2 4 Ï}

9 p

4 Ï}

9 }

4 Ï}

9 5

3 Ï}

36 } 2

5 3

4 Ï}

9 }

2 4 Ï}

81

5 3 Ï

}

3 } 6

5 Ï

}

3 } 2

31. 3 Ï}

9 }

5 Ï}

27 p

5 Ï}

9 }

5 Ï}

9 5

91/3 p 91/5

} 5 Ï}

243

5 9(1/3 1 1/5)

} 3

5 98/15

} 3

5 (91/2)16/15

} 3

5 316/15

} 3

5 31/15

5 15

Ï}

3

32. 2 6 Ï}

3 1 7 6 Ï}

3 5 (2 1 7) 6 Ï}

3 5 9 6 Ï}

3

33. 3 } 5 3 Ï}

5 2 1 } 5 3 Ï}

5 5 1 3 } 5 2 1 } 5 2 3 Ï

} 5 5

2 3 Ï}

5 } 5

34. 25 5 Ï}

2 2 15 5 Ï}

2 5 (25 2 15) 5 Ï}

2 5 10 5 Ï}

2

35. 1 }

8 4 Ï}

7 1 3 } 8 4 Ï}

7 5 1 1 } 8 1 3 } 8 2 4 Ï

} 7 5

4 Ï}

7 } 2

36. 6 3 Ï}

5 1 4 3 Ï}

625 5 6 3 Ï}

5 1 4 3 Ï}

125 p 5

5 6 3 Ï}

5 1 4 3 Ï}

125 p 3 Ï}

5

5 6 3 Ï}

5 1 4 p 5 3 Ï}

5

5 6 3 Ï}

5 1 20 3 Ï}

5

5 (6 1 20) 3 Ï}

5

5 26 3 Ï}

5

37. 26 7 Ï}

2 1 2 7 Ï}

256 5 26 7 Ï}

2 1 2 7 Ï}

128 p 2

5 26 7 Ï}

2 1 2 7 Ï}

128 p 7 Ï}

2

5 26 7 Ï}

2 1 2 p 2 p 7 Ï}

2

5 26 7 Ï}

2 1 4 7 Ï}

2

5 (26 1 4) 7 Ï}

2 5 22 7 Ï}

2

38. 12 4 Ï}

2 2 7 4 Ï}

512 5 12 4 Ï}

2 2 7 4 Ï}

256 p 2

5 12 4 Ï}

2 2 7 4 Ï}

256 p 4 Ï}

2

5 12 4 Ï}

2 2 7 p 4 4 Ï}

2

5 12 4 Ï}

2 2 28 4 Ï}

2

5 (12 2 28) 4 Ï}

2

5 216 4 Ï}

2

39. 2 4 Ï}

1250 2 8 4 Ï}

32 5 2 4 Ï}

625 p 2 2 8 4 Ï}

16 p 2

5 2 4 Ï}

625 p 4 Ï}

2 2 8 4 Ï}

16 p 4 Ï}

2

5 2 p 5 4 Ï}

2 2 8 p 2 4 Ï}

2

5 10 4 Ï}

2 2 16 4 Ï}

2

5 (10 2 16) 4 Ï}

2

5 26 4 Ï}

2

40. 5 3 Ï}

48 2 3 Ï}

750 5 5 3 Ï}

8 p 6 2 3 Ï}

125 p 6

5 5 3 Ï}

8 p 3 Ï}

6 2 3 Ï}

125 p 3 Ï}

6

5 5 p 2 p 3 Ï}

6 2 5 p 3 Ï}

6

5 10 3 Ï}

6 2 5 3 Ï}

6

5 (10 2 5) 3 Ï}

6

5 5 3 Ï}

6

41. The radical expressions 2 3 Ï}

10 and 6 3 Ï}

5 are not like radicals because they don’t have the same radicand.

Therefore, 2 3 Ï}

10 1 6 3 Ï}

5 cannot be combined.

42. To make the denominator a perfect cube, you must multiply the numerator and denominator of the fraction by y so that the value of the fraction does not change.

3 Î}

x

} y2 5 3 Î}

x p y

} y2 p y

5 3 Ï

}

xy }

3 Ï}

y3 5

3 Ï}

xy } y

43. x1/4 p x1/3 5 x(1/4 1 1/3) 5 x7/12

44. ( y4)1/6 5 y(4 p 1/6) 5 y2/3

45. 4 Ï}

81x4 5 4 Ï}

34x4 5 4 Ï}

34 p 4 Ï}

x4 5 3x

46. 2 }

x23/2 5 2x3/2

47. x2/5y

} xy21/3 5 x(2/5 2 1)y(1 1 1/3) 5 x23/5y4/3 5

y4/3

} x3/5

48. 3 Î}

x15

} y6 5

3 Ï}

x15 }

3 Ï}

y6 5

3 Ï}

(x5)3 }

3 Ï}

( y2)3 5

x5

} y2

49. ( 3 Ï}

x2 p 6 Ï}

x4 )23 5

1 }

( 3 Ï}

x2 p 6 Ï}

x4 )3 5 1 }

(x2/3 p x4/6)3

5 1 }

[x(2/3 1 4/6)]3 5

1 }

(x4/3)3 5

1 }

x4


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50. 3 Î}

x p Ï}

x5 }

Ï}

25x16 5

3 Î}

x p Ï}

x5 }

Ï}

25 p Ï}

x16 5

3 Î}

x p Ï}

x5 }

Ï}

25 p Ï}

(x8)2

5 3 Ï}

x p Ï}

x5 }

5x8 5 x1/3 p x5/2

} 5x8

5 x(1/3 1 5/2)

} 5x8 5

x17/6

} 5x8

5 x(17/6 2 8)

} 5 5 x231/6

} 5

5 1 }

5x31/6 5 1 }

5x30/6 p x1/6

5 1 }

5x5 6 Î}

x

51. Sample answer: x7/8, x1/8

52. Ï}

49x5 5 Ï}

49x4x 5 Ï}

49x4 p Ï}

x 5 7x2 Ï}

x

53. 4 Ï}

12x2y6z12 5 4 Ï}

12x2y4y2z12

5 4 Ï}

y4z12 p 4 Ï}

12x2y2

5 yz3 4 Ï}

12x2y2

54. 3 Ï}

4x3y5 p 3 Ï}

12y2 5 3 Ï}

48x3y7

5 3 Ï}

8 p 6x3y6y

5 3 Ï}

8x3y6 p 3 Ï}

6y

5 2xy2 3 Ï}

6y

55. Ï}

x2yz3 p Ï}

x3z5 5 Ï}

x5yz8

5 Ï}

x4xyz8

5 Ï}

x4z8 p Ï}

xy

5 x2z4 Ï}

xy

56. 23

} 5 Ï}

x6 5

23 }

5 Ï}

x6 p

5 Ï}

x4 }

5 Ï}

x4 5

23 5 Ï}

x4 }

5 Ï}

x10 5

23 5 Ï}

x4 }

x2

57. 3 Î}

x3

} y4 5 3 Î}

x3 p y2

} y4 p y2 5

3 Ï}

x3y2 }

3 Ï}

y6 5

3 Ï}

x3 p 3 Ï}

y2 }

3 Ï}

y6 5

x 3 Ï}

y2 }

y2

58. Î} 20x3y2

} 9xz3 5 Î}

20x3y2 p xz }

9xz3 p xz 5 Î}

20x4y2z }

9x2z4

5 Ï}

4 p 5x4y2z }

Ï}

9x2z4 5

Ï}

4x4y2 p Ï}

5z }

Ï}

9x2z4

5 2x2y Ï

} 5z }

3xz2 5 2xy Ï

} 5z }

3z2

59. 4 Ï}

x6 }

7 Ï}

x5 5

4 Ï}

x6 }

7 Ï}

x5 p

7 Ï}

x2 }

7 Ï}

x2 5

4 Ï}

x6 p 7 Ï}

x2 }

7 Ï}

x7

5 x6/4 p x2/7

} x 5 x(6/4 1 2/7)

} x

5 x25/14

} x 5 x(25/14 2 1) 5 x11/14 5 14

Ï}

x11

60. 3 5 Ï}

x 1 9 5 Ï}

x 5 (3 1 9) 5 Ï}

x 5 12 5 Ï}

x

61. 3 }

4 y3/2 2

1 } 4 y3/2 5 1 3 } 4 2

1 } 4 2 y3/2 5

y3/2

} 2

62. 27 3 Î}

y 1 16 3 Î}

y 5 (27 1 16) 3 Î}

y 5 9 3 Î}

y

63. (x4y)1/2 1 (xy1/4)2 5 x(4 p 1/2)y1/2 1 x2y(1/4 p 2)

5 x2y1/2 1 x2y1/2 5 2x2y1/2

64. x Ï}

9x3 2 2 Ï}

x5 5 x Ï}

9x2x 2 2 Ï}

x4x

5 3x2 Ï}

x 2 2x2 Ï}

x

5 (3 2 2)x2 Ï}

x

5 x2 Ï}

x

65. y 4 Ï}

32x6 1 4 Ï}

162x2y4 5 y 4 Ï}

16 p 2x4x2 1 4 Ï}

81 p 2x2y4

5 2xy 4 Ï}

2x2 1 3y 4 Ï}

2x2

5 (2xy 1 3y) 4 Ï}

2x2

66. P 5 2l 1 2w A 5 lw

5 2(x3) 1 2(2x2/3) 5 x3(2x2/3) 5 2x3 1 4x2/3 5 2x(3 1 2/3)

5 2x11/3

67. P 5 2l 1 2w A 5 lw

5 2(7x1/4) 1 2(5x1/4) 5 (7x1/4)(5x1/4) 5 14x1/4 1 10x1/4 5 35x(1/4 1 1/4)

5 (14 1 10)x1/4 5 35x1/2

5 24x1/4

68. c2 5 a2 1 b2

c2 5 (3x1/3)2 1 (4x1/3)2

c2 5 9x2/3 1 16x2/3

c2 5 25x2/3

c 5 Ï}

25x2/3

c 5 5 Ï}

(x1/3)2

c 5 5x1/3

P 5 a 1 b 1 c A 5 1 } 2 bh

5 3x1/3 1 4x1/3 1 5x1/3 5 1 } 2 (4x1/3)(3x1/3)

5 12x1/3 5 1 } 2 p 12x(1/3 1 1/3)

5 6x2/3

69. C; 2 1 } 6 Ï

}

4x 2 1 } 6 Ï

}

9x 5 2 1 } 6 p 2 Ï

}

x 2 1 } 6 p 3 Ï

}

x

5 1 2 1 } 3 2

1 } 2 2 Ï}

x

5 2 5 } 6 Ï

}

x

70. x0.5 p x2 5 x(0.5 1 2) 5 x2.5

71. y20.6 p y26 5 y[20.6 1 (26)] 5 y26.6 5 1 }

y6.6

72. (x6y2)20.75 5 x6 p (20.75)y2 p (20.75)

5 x24.5y21.5 5 1 }

x4.5y1.5

73. x0.3

} x1.5 5 x(0.3 2 1.5) 5 x21.2 5

1 }

x1.2

74. (x5y23)20.25 5 x5 p (20.25)y23 p (20.25)

5 x21.25y0.75 5 y0.75

} x1.25

75. y20.5

} y0.8 5 y(20.5 2 0.8) 5 y21.3 5

1 }

y1.3

76. 10x0.6 1 (4x0.3)2 5 10x0.6 1 16x0.3 p 2

5 10x0.6 1 16x0.6 5 26x0.6


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369Algebra 2


77. 15z0.3 2 (2z0.1)3 5 15z0.3 2 8z0.3 5 7z0.3

78. x5 Ï

}

3 }

x2 Ï}

3 5 x(5 Ï

}

3 2 2 Ï}

3 ) 5 x3 Ï}

3

79. (x Ï}

2 ) Ï}

3 5 x( Ï}

2 p Ï}

3 ) 5 x Ï}

6

80. 1 xπ }

xπ/3 2 2 5

x2π }

x2π/3 5 x(2π 2 2π/3) 5 x4π/3

81. x2y Ï}

2 1 3x2y Ï}

2 5 4x2y Ï}

2

82. a. 3 }

9x 5 243 b. 2x p 2x 1 1 5 1 } 16

3 5 243 p 9x 2x 1 (x 1 1) 5 1 } 16

3 }

243 5 9x 22x 1 1 5

1 } 16

1 }

81 5 9x 22x 1 1 5

1 }

24

1 }

92 5 9x 22x 1 1 5 224

922 5 9x 2x 1 1 5 24

x 5 22 2x 5 25

x 5 2 5 } 2

c. (4x)x 1 2 5 64

4x p (x 1 2) 5 64

4x2 1 2x 5 64

4x2 1 2x 5 43

x2 1 2x 5 3

x2 1 2x 2 3 5 0

(x 1 3)(x 2 1) 5 0

x 5 23 or x 5 1

Problem Solving

83. a. S 5 km2/3 b. S 5 km2/3

5 57.5(32)2/3 5 11(5.9 3 104)2/3

ø 57.5(10.08) 5 11(5.9)2/3(104)2/3

ø 580 ø 11(3.27)(108/3) The bat’s surface area ø 11(3.27)(464.16)

is about 580 square ø 16,696

centimeters. The human’s surface

area is about 16,700

square centimeters.

84. v 5 8.8 Î}

L

} A

5 8.8 Î} 1.4 3 107

} 5.5 3 103

ø 8.8 Ï}

0.25 3 104

5 8.8 Ï}

2500 5 8.8(50) 5 440

The velocity of the jet is about 440 feet per second.

85. d 5 1.9[ 1 5.5 3 1024 2 l]1/2 l 5 10 cm 5 100 mm

5 1.9[(5.5 3 1024)(100)]1/2

5 1.9(0.055)1/2

ø 1.9(0.23) 5 0.44

The optimum pinhole diameter is about 0.44 millimeter.

86.

x

xx 2

a2 1 b2 5 c2

x2 1 x2 0 (x Ï}

2 )2

2x2 0 x2 p ( Ï}

2 )2

2x2 5 2x2 ✓

87. a. 2.512m1

} 2.512m2

5 2.5120.77

} 2.5120.03 5 2.5120.74 ø 1.98

Altair is about 2 times fainter than Vega.

b. 2.512m1

} 2.512m2

5 2.5121.25

} 2.5120.77 5 2.5120.48 ø 1.56

Deneb is about 1.6 times fainter than Altair.

c. 2.512m1

} 2.512m2

5 2.5121.25

} 2.5120.03 5 2.5121.22 ø 3.08

Deneb is about 3 times fainter than Vega.

88. d 5 V0 Ï

}}

(V0)2 1 2gh0

}} g 5 V0 Ï

}}

(V0)2 1 2g(0)

}} g

5 V0 Ï

}

(V0)2 } g 5

V0 p V0 } g 5

(V0)2

} g

89. a. V 5 4 } 3 πr3

3V

} 4π� 5 r3

3 Î}

3V

} 4π� 5 r

b. S 5 4πr2

5 4π 1 3 Î}

3V

} 4π� 2

2

5 4π 1 3V }

4π� 2 2/3

5 4π (3V)2/3

} (4π)2/3

5 (4π)(1 2 2/3)(3V)2/3

5 (4π)1/3(3V)2/3

c. Balloon with volume V:

S1 5 (4π)1/3(3V)2/3

Balloon with volume 2V:

S2 5 (4π)1/3(3 p 2V)2/3 5 (4π)1/3(2)2/3(3V)2/3

5 (4π)1/3(4)1/3(3V)2/3

5 3 Ï}

4 (4π)1/3(3V)2/3 5 3 Ï}

4 S1

The balloon with twice as much water has 3 Ï}

4 , or about 1.59 times the surface area of the other balloon.


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90. Sample answer: n Ï}

xm

n 5 3, m 5 5, x 5 21: 3 Ï}

(21)5 5 21

n 5 4, m 5 5, x 5 21: 4 Ï}

(21)5 5 4 Ï}

21 , but 21 has no

real 4th root; 4 Ï}

(21)5 5 1

n 5 3, m 5 4, x 5 21: 3 Ï}

(21)4 5 1

n 5 2, m 5 4, x 5 21: Ï}

(21)4 5 1

Absolute value is needed when n is even and m is odd.

Mixed Review

91. x 2 7 ≥ 15

x ≥ 22

92. 10x 1 7 < 24x 1 9 93. 3x ≤ 26x 2 20

14x 1 7 < 9 9x ≤ 220

14x < 2 x ≤ 2 20

} 9

x < 1 } 7

94. x2 1 7x 1 10 > 0

x2 1 7x 1 10 5 0

(x 1 5)(x 1 2) 5 0

x 5 25 or x 5 22

2 2102223 212426 25

Test x = 26:(26)2 1 7(26) 1 10 0 4 0

Test x = 23:(23)2 1 7(23) 1 10 0 22 0²

,,

?

,?

Test x = 0: (0)2 1 7(0) 1 10 0 10 0

,,

?

The solution is x < 25 or x > 22.

95. 2x2 1 4x ≥ 232 0 ≥ x2 2 4x 2 32

0 5 x2 2 4x 2 32

0 5 (x 2 8)(x 1 4)

x 5 8 or x 5 24

4 6 8

9

202226 24

25

Test x = 25:2(25)2 1 4(25) 232 245 232£

? Test x = 0: 202 1 4(0) 232 0 232£

? Test x = 9: 292 1 4(9) 232 245 232£

?

10 12

The solution is 24 ≤ x ≤ 8.

96. 6x2 1 x 2 7 < 5

6x2 1 x 2 12 < 0

6x2 1 x 2 12 5 0

(3x 2 4)(2x 1 3) 5 0

x 5 4 } 3 or x 5 2

3 } 2

1 2 30212223

Test x = 22:6(22)2 2 2 2 7 5 15 5² ²

,? Test x = 0: 6(0)2 1 0 2 7 5 27 5

,,

? Test x = 2: 6(2)2 1 2 2 7 5 19 5

,?

43

32

The solution is 2 3 } 2 < x <

4 }

3 .

97. 3 1 22 21 23

3 3 6

1 1 2 3

f (3) 5 3

98. 23 1 22 21 23

23 15 242

1 25 14 245

f (23) 5 245

99. 5 1 22 21 23

5 15 70

1 3 14 67

f (5) 5 67

100. 24 1 22 21 23

24 24 292

1 26 23 295

f (24) 5 295

101. (12x2 1 2x) 2 (28x3 1 5x2 2 9x) 5 12x2 1 2x 1 8x3 2 5x2 1 9x

5 8x3 1 7x2 1 11x

102. (35x3 2 14) 1 (215x2 1 7x 1 20) 5 35x3 2 15x2 1 7x 2 14 1 20

5 35x3 2 15x2 1 7x 1 6

103. 18x2(x 1 4) 5 18x2(x) 1 18x2(4)

5 18x3 1 72x2

104. (8x 2 3)2 5 (8x)2 2 2(8x)(3) 1 (23)2

5 64x2 2 48x 1 9

105. (x 2 4)(x 1 1)(x 1 2)

5 (x2 2 3x 2 4)(x 1 2)

5 (x2 2 3x 2 4)x 1 (x2 2 3x 2 4)2 5 x3 2 3x2 2 4x 1 2x2 2 6x 2 8

5 x3 2 x2 2 10x 2 8

106. 3 1 1 27 215

3 12 15

1 4 5 0

x3 1 x2 2 7x 2 15

}} x 2 3

5 x2 1 4x 1 5


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371Algebra 2


Quiz 6.1–6.2 (p. 427)

1. 363/2 5 ( Ï}

36 )3 5 63 5 216

2. 6422/3 5 1 }

642/3 5 1 }

( 3 Ï}

64 )2 5

1 }

42 5 1 } 16

3. 2(6253/4) 5 2[( 4 Ï}

625 )3] 5 2(53) 5 2125

4. (232)2/5 5 ( 5 Ï}

232 )2 5 (22)2 5 4

5. x4 5 20 6. x5 5 210

x 5 6 4 Ï}

20 x 5 5 Ï}

210

x ø 62.11 x ø 21.58

7. x6 1 5 5 26 8. (x 1 3)3 5 216

x6 5 21 x 1 3 5 3 Ï}

216

x 5 6 6 Î}

21 x 5 3 Ï}

216 2 3

x ø 61.66 x ø 25.52

9. 4 Ï}

32 p 4 Ï}

8 5 4 Ï}

32 p 8 5 4 Ï}

256 5 4

10. ( Ï}

10 p 3 Ï}

10 )8 5 (101/2 p 101/3)8

5 F 10(1/2 1 1/3) G 8 5 (105/6)8

5 1020/3

5 3 Ï}

1020

5 3 Ï}

1018 p 102

5 3 Ï}

(106)3 p 102

5 106 3 Ï}

102

5 1,000,000 3 Ï}

100

11. (x6y4)1/8 1 2(x1/3y1/4)2 5 x3/4y1/2 1 2(x2/3y1/2) 5 x3/4y1/2 1 2x2/3y1/2

12. 3 Ï

}

73 1 4 Ï}

73 }}

Ï}

75 5

7 Ï}

73 }

Ï}

75 p Ï

}

75 }

Ï}

75

5 7 p 73/2 p 75/2

} 75 5

71 1 3/2 1 5/2

} 75 5

75

} 75 5 1

13. 2 Ï

}

x p Ï}

x3 }

Ï}

64x15 5

2 Ï}

x4 }

Ï}

82 p x14 p x 5

2x2

} 8x7 Ï}

x p Ï

}

x }

Ï}

x 5

2x2 Ï} x }

8x8 5 Ï}

x }

4x6

14. y2 5 Ï}

64x6 2 6 5 Ï}

2x6y10 5 y2 5 Ï}

32 p 2x5x 2 6 5 Ï}

2x5xy10

5 2xy2 5 Ï}

2x 2 6xy2 5 Ï}

2x

5 24xy2 5 Ï}

2x

15. Three labels A, B, C, are added to the graph to indicate the three right triangles.

4

4

8

2

AC

B

For right triangle A

a2 1 b2 5 c2

(8 2 2)2 1 82 5 c2

62 1 82 5 c2

100 5 c2

10 5 c

For right triangle B For right triangle C

a2 1 b2 5 c2 a2 1 b2 5 c2

42 1 82 5 c2 22 1 42 5 c2

80 5 c2 20 5 c2

4 Ï}

5 5 c 2 Ï}

5 5 c

The perimeter of the right triangle is 10 1 4 Ï

} 5 1 2 Ï

} 5 5 10 1 6 Ï

} 5 .

Lesson 6.3


1. f (x) 1 g(x) 5 22x2/3 1 7x2/3 5 (22 1 7)x2/3 5 5x2/3

2. f (x) 2 g(x) 5 22x2/3 2 7x2/3 5 (22 2 7)x2/3 5 29x2/3

3. The functions f and g each have the same domain: all real numbers. So, the domains of f 1 g and f 2 g also consist of all real numbers.

4. f (x) p g(x) 5 3x(x1/5) 5 3x(1 1 1/5) 5 3x6/5

5. f (x)

} g(x)

5 3x

} x1/5 5 3x(1 2 1/5) 5 3x4/5

6. The functions f and g each have the same domain: all real numbers. So, the domain of f p g also consists of all real

numbers. Because g(0) 5 0, the domain of f } g is restricted

to all real numbers except x 5 0.

7. r(m) p s(m) 5 (1.446 3 109)m20.05

5 (1.446 3 109)(1.7 3 105)20.05

ø (1.446 3 109)(0.55)

ø 791,855,335 The white rhino has about 791,855,335 heartbeats over

its lifetime.

8. g( f (5)) 9. f (g(5))

f(5) 5 3(5) 2 8 5 7 g(5) 5 2(5)2 5 50

g( f (5)) 5 g(7) f (g(5)) 5 f (50)

5 2(7)2 5 3(50) 2 8

5 98 5 142

10. f ( f (5)) 11. g(g (5))

f (5) 5 3(5) 2 8 5 7 g(5) 5 2(5)2 5 50

f ( f (5)) 5 f (7) g(g(5)) 5 g(50)

5 3(7) 2 8 5 2(50)2

5 13 5 5000

12. f (x) 5 2x21, g(x) 5 2x 1 7

f (g(x)) 5 f (2x 1 7) 5 2(2x 1 7)21 5 2 } 2x 1 7

g( f (x)) 5 g(2x21) 5 2(2x21) 1 7 5 4x21 1 7 5 4 } x 1 7

f( f (x)) 5 f (2x21) 5 2(2x21)21 5 2(221x) 5 20x 5 x

The domain of f (g(x)) consists of all real numbers except

x 5 2 7 } 2 because g 1 2

7 } 2 2 5 0 is not in the domain of f.

The domains of g( f (x)) and f ( f (x)) consist of all real numbers except x 5 0, again because 0 is not in the domain of f.

13. Function for $15 gift certifi cate: f (x) 5 x 2 15 Function for 20% discount: g(x) 5 x 2 0.2x 5 0.8x

g( f (x)) 5 g(x 2 15) 5 0.8(x 2 15)

f (g(x)) 5 f (0.8x) 5 0.8x 2 15

When x 5 55:

g( f (55)) 5 0.8(55 2 15) 5 0.8(40) 5 $32

f (g(55)) 5 0.8(55) 2 15 5 44 2 15 5 $29

The sale price is $32 when the $15 gift certifi cate is applied before the 20% discount. The sale price is $29 when the 20% discount is applied before the $15 gift certifi cate.


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Skill Practice

1. The function h(x) 5 g( f (x)) is called the composition of the function g with the function f.

2. The sum of two power functions is sometimes a power function.

Sample answer:

f (x) 5 2x1/3, g(x) 5 4x21/3

f (x) 1 g(x) 5 2x1/3 1 4x21/3

f (x) 5 2x1/3, g(x) 5 4x1/3

f (x) 1 g(x) 5 2x1/3 1 4x1/3 5 6x1/3

3. f (x) 1 g(x) 5 23x1/3 1 4x1/2 1 5x1/3 1 4x1/2

5 (23 1 5)x1/3 1 (4 1 4)x1/2

5 2x1/3 1 8x1/2

Domain: all nonnegative real numbers

4. g(x) 1 f (x) 5 5x1/3 1 4x1/2 2 3x1/3 1 4x1/2

5 (5 2 3)x1/3 1 (4 1 4)x1/2

5 2x1/3 1 8x1/2


5. f (x) 1 f (x) 5 23x1/3 1 4x1/2 1 (23x1/3 1 4x1/2) 5 (23 2 3)x1/3 1 (4 1 4)x1/2

5 26x1/3 1 8x1/2


6. g(x) 1 g(x) 5 5x1/3 1 4x1/2 1 5x1/3 1 4x1/2

5 (5 1 5)x1/3 1 (4 1 4)x1/2

5 10x1/3 1 8x1/2


7. f (x) 2 g(x) 5 23x1/3 1 4x1/2 2 (5x1/3 1 4x1/2) 5 (23 2 5)x1/3 1 (4 2 4)x1/2

5 28x1/3


8. g(x) 2 f (x) 5 5x1/3 1 4x1/2 2 (23x1/3 1 4x1/2) 5 (5 1 3)x1/3 1 (4 2 4)x1/2

5 8x1/3


9. f (x) 2 f (x) 5 23x1/3 1 4x1/2 2 (23x1/3 1 4x1/2) 5 (23 1 3)x1/3 1 (4 2 4)x1/2

5 0


10. g(x) 2 g(x) 5 5x1/3 1 4x1/2 2 (5x1/3 1 4x1/2) 5 (5 2 5)x1/3 1 (4 2 4)x1/2

5 0


11. B;

f (x) 1 g(x) 5 27x2/3 2 1 1 2x2/3 1 6

5 (27 1 2)x2/3 2 1 1 6

5 25x2/3 1 5

12. f (x) p g(x) 5 4x2/3 p 5x1/2

5 20x(2/3 1 1/2)

5 20x7/16

Domain of f: all real numbers

Domain of g: all nonnegative real numbers

Domain of f p g: all nonnegative real numbers

13. g(x) p f(x) 5 5x1/2 p 4x2/3

5 20x(1/2 1 2/3)

5 20x7/6



Domain of g p f: all nonnegative real numbers

14. f (x) p f(x) 5 4x2/3 p 4x2/3

5 16x(2/3 1 2/3)

5 16x4/3


Domain of f p f: all real numbers

15. g(x) p g(x) 5 5x1/2 p 5x1/2

5 25x(1/2 1 1/2)

5 25x


Domain of g p g: all nonnegative real numbers

16. f (x)

} g(x)

5 4x2/3

} 5x1/2 5

4x(2/3 2 1/2)

} 5 5 4x1/6

} 5



Domain of f }

g : all positive real numbers

17. g(x)

} f (x)

5 5x1/2

} 4x2/3 5

5x(1/2 2 2/3)

} 4 5 5x21/6

} 4 5 5 }

4x1/6



Domain of g }

f : all positive real numbers

18. f (x)

} f (x)

5 4x2/3

} 4x2/3 5 1


Domain of f }

f : all real numbers except x 5 0

19. g(x)

} g(x)

5 5x1/2

} 5x1/2 5 1


Domain of g }

g : all positive real numbers

20. g(23) 5 2(23)2 5 29

f (g(23)) 5 f (29) 5 3(29) 1 2 5 225

21. f (2) 5 3(2) 1 2 5 8

g( f (2)) 5 g(8) 5 282 5 264


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373Algebra 2


22. f (29) 5 3(29) 1 2 5 225

h( f (29)) 5 h(225) 5 225 2 2

} 5 5 2 27

} 5

23. h(8) 5 8 2 2

} 5 5 6 } 5

g(h(8)) 5 g 1 6 } 5 2 5 2 1 6 } 5 2 2 5 2 36

} 25

24. g(5) 5 252 5 225

h(g(5)) 5 h(225) 5 225 2 2

} 5 5 2 27

} 5

25. f (7) 5 3(7) 1 2 5 23

f ( f (7)) 5 f (23) 5 3(23) 1 2 5 71

26. h(24) 5 24 2 2

} 5 5 2 6 } 5

h(h(24)) 5 h 1 2 6 } 5 2 5

2 6 } 5 2 2 } 5 5 2

16 } 25

27. g(25) 5 2(25)2 5 225

g(g(25)) 5 g(225) 5 2(225)2 5 2625

28. f (g(x)) 5 f (2x 2 7) 5 3(2x 2 7)21 5 3 } 2x 2 7

The domain of f (g(x)) consists of all real numbers except

x 5 7 } 2 because g 1 7 }

2 2 5 0 is not in the domain of f.

29. g( f (x)) 5 g(3x21) 5 2(3x21) 2 7 5 6x21 2 7 5 6 } x 2 7

The domain of g( f (x)) consists of all real numbers except x 5 0 because 0 is not in the domain of f.

30. h( f (x)) 5 h(3x21) 5 3x21 1 4

} 3 5 3x21

} 3 1 4 } 3 5

1 } x 1

4 } 3

The domain of h( f (x)) consists of all real numbers except x 5 0 because 0 is not in the domain of f.

31. g(h(x)) 5 g 1 x 1 4 }

3 2 5 2 1 x 1 4

} 3 2 2 7

5 2x 1 8

} 3 2 7 5 2x 1 8 2 21

} 3 5 2x 2 13

} 3

The domain of g(h(x)) consists of all real numbers.

32. h(g(x)) 5 h(2x 2 7) 5 2x 2 7 1 4

} 3 5 2x 2 3

} 3

The domain of h(g(x)) consists of all real numbers.

33. f ( f (x)) 5 f (3x21) 5 3(3x21)21 5 3(321x) 5 30x 5 x

The domain of f ( f (x)) consists of all real numbers except x 5 0, because 0 is not in the domain of f.

34. h(h(x)) 5 h 1 x 1 4 }

3 2 5

x 1 4

} 3 1 4 } 3 5

x 1 4 1 12 } 9 5

x 1 16 } 9

The domain of h(h(x)) consists of all real numbers.

35. g(g(x)) 5 g(2x 2 7) 5 2(2x 2 7) 2 7

5 4x 2 14 2 7 5 4x 2 21

The domain of g(g(x)) consists of all real numbers.

36. When performing f (4x), 4x should have been substituted for x in the function f.

f (g(x)) 5 f (4x) 5 (4x)2 2 3 5 16x2 2 3

37. The product 4(x2 2 3) was not performed correctly.

g( f (x)) 5 g(x2 2 3) 5 4(x2 2 3) 5 4x2 2 12

38. A;

g( f (x)) 5 g(7x2) 5 3(7x2)22 5 3(722x24) 5 3 }

49x4

39. Sample answer: f (x) 5 x, g(x) 5 x21

f (g(x)) 5 g( f (x))

f (x21) 5 g(x)

x21 5 x21

40. Sample answer:

f (x) 5 3 Ï}

x , g(x) 5 x 1 2

h(x) 5 f (g(x)) 5 f (x 1 2) 5 3 Ï}

x 1 2

41. Sample answer:

f (x) 5 4 } x 1 7 , g(x) 5 3x2

h(x) 5 f (g(x)) 5 f (3x2) 5 4 }

3x2 1 7

42. Sample answer:

f (x) 5 x, g(x) 5 2x 1 9

h(x) 5 f (g(x)) 5 f (2x 1 9) 5 2x 1 9

Problem Solving

43. r(w) 5 1.1w0.734

} b(w) 2 d(w)

5 1.1w0.734

}} 0.007w 2 0.002w

5 1.1w0.734

} 0.005w

5 220w(0.734 2 1)

5 220w20.266

r(w) 5 220w20.266

r(6.5) 5 220(6.5)20.266 ø 134

The breathing rate of a mammal that weighs 6.5 grams is about 134 breaths per minute.

r(300) 5 220(300)20.266 ø 48.3

The breathing rate of a mammal that weighs 300 grams is about 48.3 breaths per minute.

r (70,000) 5 220(70,000)20.266 ø 11.3

The breathing rate of a mammal that weighs 70,000 grams is about 11.3 breaths per minute.

44. C(x(t)) 5 C(50t) 5 60(50t) 1 750 5 3000t 1 750

C(x(5)) 5 3000(5) 1 750 5 15,750

This number represents the cost ($15,750) of 5 hours of production in the factory.

45. Let x represent the regular price.

Function for $15 discount: f (x) 5 x 2 15

Function for 10% discount: g(x) 5 x 2 0.1x 5 0.9x

a. g( f (x)) 5 g(x 2 15) 5 0.9(x 2 15)

g( f (85)) 5 0.9(85 2 15) 5 0.9(70) 5 63

The sale price is $63 when the $15 discount is applied before the 10% discount.

b. f (g(x)) 5 f (0.9x) 5 0.9x 2 15

f (g(85)) 5 0.9(85) 2 15 5 76.5 2 15 5 61.50

The sale price is $61.50 when the 10% discount is applied before the $15 discount.


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c. If the 10% discount is applied before the $15 discount, you get a better deal. Your purchase will be $61.50 instead of $63.

46. a. Distance from point A to point D: 20 2 x

Distance 5 rate p time

20 2 x 5 (6.4)r(x)

20 2 x

} 6.4

5 r (x)

Distance from point D to point B:

x2 1 122 5 c2

x2 1 144 5 c2

Ï}

x2 1 144 5 c

Distance 5 rate p time

Ï}

x2 1 144 5 (0.9)s(x)

Ï}

x2 1 144 }

0.9 5 s(x)

b. t(x) 5 r(x) 1 s(x) 5 20 2 x

} 6.4 1 Ï}

x2 1 144 } 0.9

c.

MinimumX=1.7044344 Y=16.325839

The value of x that minimizes t(x) is 1.7. This means that to get to the ball in the shortest time, Elvis should run along the beach 20 2 1.7 5 18.3 meters and then swim out to the ball.

47. a. f (1) 5 1 1

2 } 1 } 2 5

3 } 2 5 1.5

f ( f (1)) 5 f (1.5) 5 1.5 1

2 } 1.5 } 2 ø 1.417

f ( f ( f (1))) 5 f (1.417) 5 1.417 1

2 } 1.417 }} 2 ø 1.414

f( f ( f ( f (1)))) 5 f (1.414)

5 1.414 1

2 } 1.414 }} 2 ø 1.414214

b. f ( f ( f ( f ( f (1))))) 5 f(1.414214)

5 1.414214 1

2 } 1.414214 }} 2 ø 1.414214

Ï}

2 ø 1.414213562

You need to compose the function 3 times in order for the result to approximate Ï

}

2 to three decimal places. You need to compose the function 4 times in order for the result to approximate Ï

}

2 to six decimal places.

Mixed Review

48. y 2 2x 5 12 49. 3x 2 2y 5 10

y 5 2x 1 12 22y 5 23x 1 10

y 5 3 } 2 x 2 5

50. x 5 23y 1 9 51. 3x 2 4y 5 7

3y 5 2x 1 9 24y 5 23x 1 7

y 5 2 1 } 3 x 1 3 y 5

3 } 4 x 2

7 } 4

52. x 2 y 5 12 53. ax 1 by 5 c

2y 5 2x 1 12 by 5 2ax 1 c

y 5 x 2 12 y 5 2 a }

b x 1

c }

b

54.

2

x

y

22

55.

2

x

y

22

56.

1

x

y

21

57.

2

x

y

22

58. 1

x

y

21

59.

4

x

y

21

60. 1

x

y

1

61.

2

x

y

21

62.

1

x

y

22

63. 5

x

y

21

64.

1

x

y

21

65.

1

x

y

21


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375Algebra 2


66.1

x

y

1

Graphing Calculator Activity 6.3 (p. 435)

1.Y1=X3+5X-3Y2=-3X2-XY3=Y2+Y1Y4=Y5=Y6=Y7=Y7=

Y3(7) 221

2.Y1=X̂ (1/3)

Y2=9XY3=Y1/Y2Y4=Y5=Y6=Y7=Y7=

Y3(-8) 0.0277777778

3.Y1=5X3-3X2Y2=-2X2-5Y3=Y2-Y1Y4=Y5=Y6=Y7=Y7=

Y3(2) -41

4.Y1=2X2+7X-2Y2=X-6Y3=Y1(Y2)Y4=Y5=Y6=Y7=Y7=

Y3(5) -7

Mixed Review of Problem Solving (p. 436)

1. a. s(x) 5 x2

b. c(x) 5 π 1 x } 2 2 2 5 π x2

} 4

c. r(x) 5 x2 2 π x2

} 4 5 1 1 2 π

} 4 2 x2 ø 0.21x2

2. a. V 5 321(4π)21/2(S3)1/2

5 S3/2

} 3(2π1/2)

5 S Ï

}

S }

6π1/2

5 S

} 6π� Ï}

πS

b. V 5 79

} 6π Ï}

π(79) ø 66

The volume of the candlepin bowling ball is about 66 cubic inches.

c. V 5 232

} 6π Ï}

π(232) ø 332

The volume of the 10-pin bowling ball is about 332 cubic inches.

d. The surface area of the 10-pin bowling ball is about 3 times that of the candlepin bowling ball. The volume of the 10-pin bowling ball is about 5 times that of the candlepin bowling pin.

3. f (x) 5 x 2 100,000, g(x) 5 0.03x

f (g(x)) 5 0.03x 2 100,000

g( f (x)) 5 0.03(x 2 100,000)

The composition g( f (x)) represents your bonus if x > 100,000, because the bonus must be applied after $100,000 is subtracted.

4. a. V 5 πr2h 5 3.14x2(5) 5 15.7x2

b. 15.7x2

} 2 5 128(8.8)

15.7x2 5 2252.8

c. 15.7x2 5 2252.8

x2 ø 143.49

x ø 6 Ï}

143.49

x ø 612

The radius of the pool is about 12 feet.

d. 4 } 5 2

1 } 2 5

3 } 10

Volume fi lled

5 Hose output per hour

p Time

3 }

10 (2252.8) 5 (104 1 128) p t

675.84 5 232t

2.9 ø t It will take a total of 8.8 1 2.9, or 11.7 hours to

fi ll 4 } 5 of the pool.

5. Sample answer:

f(x) 5 5 f(x) 5 x

f( f(x)) 5 f(5) 5 5 f( f(x)) 5 f(x) 5 x

6. Sample answer:

1 161/2 }

41/2 2 5 5 1 4 } 2 2 5 Evaluate 161/2 and 41/2.

5 25 Divide 4 by 2.

5 32 Evaluate 25.

Yes, there is another set of steps you could use to simplify the expression. For example:

1 161/2 }

41/2 2 5 5 F 1 16

} 4 2 1/2 G 5 Power of a quotient property

5 (41/2)5 Divide 16 by 4.

5 4(1/2 p 5) Power of a power property

5 45/2 Multiply 1 }

2 by 5.

5 32 Evaluate 45/2.


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7. V 5 4 } 3 πr3

900 5 4 } 3 (3.14)r3

900 ø 4.19r3

214.8 ø r3

6.0 ø r The radius of the sphere is about 6.0 inches.

Lesson 6.4

Investigating Algebra Activity 6.4 (p. 437)

1. a. f (x) 5 3x 1 2 x 22 21 0 1 2

y 5 f(x) 24 21 2 5 8

(2, 0), (5, 1)

x

y

6

6

f

g

(24, 22)

(22, 24) (21, 21)

(2, 0)

(1, 5)

(2, 8)

(8, 2)(5, 1)(0, 2)

m 5 1 2 0

} 5 2 2 5 1 } 3

y 2 y1 5 m(x 2 x1)

y 2 0 5 1 } 3 (x 2 2)

y 5 x 2 2

} 3

g(x) 5 x 2 2

} 3

2. a. You can graph the inverse of a function by refl ecting it in the line y 5 x.

3. a. In words, g is the function that subtracts 2 from x then divides the result by 3.

f (g(x)) 5 f 1 x 2 2 }

3 2 g( f (x)) 5 g(3x 1 2)

5 3 1 x 2 2 }

3 2 1 2 5

3x 1 2 2 2 } 3

5 x 2 2 1 2 5 3x

} 3

5 x 5 x

If f (g(x)) 5 x and g( f (x)) 5 x, then the function is indeed the inverse of the original function.

1. b. f (x) 5 x 2 1

} 6

x 25 22 1 4 7

y 5 f(x) 21 2 1 } 2 0

1 }

2 1

(0, 1), (1, 7)

x

y

6

6

(25, 21)

(21, 25)

(1, 0)

(1, 7)

(7, 1)(1, 0)

(22, 2 )12

(2 , 22)12

( , 4)12

(4, )12

f

g

m 5 7 2 1

} 1 2 0 5 6

y 2 y1 5 m(x 2 x1)

y 2 1 5 6(x 2 0)

y 2 1 5 6x

y 5 6x 1 1

g(x) 5 6x 1 1

2. b. You can graph the inverse of a function by refl ecting it in the line y 5 x.

3. b. In words, g is the function that multiplies x by 6 and then adds 1.

f (g(x)) 5 f (6x 1 1) g( f (x)) 5 g 1 x 2 1 }

6 2

5 6x 1 1 2 1

} 6 5 6 1 x 2 1 }

6 2 1 1

5 6x

} 6 5 x 2 1 1 1

5 x 5 x


1. c. f (x) 5 4 2 3 } 2 x x 22 0 2 4 6

y 5 f(x) 7 4 1 22 25

(4, 0), (1, 2)

x

y

1

1

(22, 7)

(4, 22)

(0, 4)

(2, 1)(1, 2)

(4, 0)

(7, 22)

(6, 25)

(25, 6)(22, 4)

g

f

m 5 2 2 0

} 1 2 4 5 2 2 } 3

y 2 y1 5 m(x 2 x1)

y 2 0 5 2 2 } 3 (x 2 4)

y 5 22(x 2 4)

} 3

y 5 22x 1 8

} 3

g(x) 5 22x 1 8

} 3

2. c. You can graph the inverse of a function by refl ecting it in the line y 5 x.

3. c. In words, g is the function that multiplies x by 22 and then adds 8, dividing the result by 3.

f (g(x)) 5 f 1 22x 1 8 }

3 2 g( f (x)) 5 g 1 4 2

3 } 2 x 2

5 4 2 3 } 2 1 22x 1 8

} 3 2 5

22 1 4 2 3 } 2 x 2 1 8

}} 3

5 4 2 1 22x 1 8 } 2 2 5

28 1 3x 1 8 } 3

5 4 2 (2x 1 4) 5 3x

} 3

5 4 1 x 2 4 5 x

5 x



1. f (x) 5 x 1 4

y 5 x 1 4

x 5 y 1 4

x 2 4 5 y

f 21(x) 5 x 2 4

f ( f 21(x)) 5 f (x 2 4) f 21( f (x)) 5 f 21(x 1 4)

5 x 2 4 1 4 5 x 1 4 2 4

5 x ✓ 5 x ✓


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377Algebra 2


2. f (x) 5 2x 2 1

y 5 2x 2 1

x 5 2y 2 1

x 1 1 5 2y

1 }

2 x 1

1 } 2 5 y

f 21(x) 5 1 } 2 x 1

1 } 2

f ( f 21(x)) 5 f 1 1 } 2 x 1

1 } 2 2 f 21( f(x)) 5 f 21(2x 2 1)

5 2 1 1 } 2 x 1

1 } 2 2 2 1 5

1 } 2 (2x 2 1) 1

1 } 2

5 x 1 1 2 1 5 x 2 1 } 2 1

1 } 2

5 x ✓ 5 x ✓

3. f (x) 5 23x 1 1

y 5 23x 1 1

x 5 23y 1 1

x 2 1 5 23y

2 1 } 3 x 1

1 } 3 5 y

f 21(x) 52 1 } 3 x 1

1 } 3

f ( f 21(x)) 5 f 1 2 1 } 3 x 1

1 } 3 2

5 23 1 2 1 } 3 x 1

1 } 3 2 1 1

5 x 2 1 1 1

5 x ✓

f 21( f(x)) 5 f 21(23x 1 1)

5 2 1 } 3 (23x 1 1) 1

1 } 3

5 x 2 1 } 3 1

1 } 3

5 x ✓

4. L 5 8 } 3 R 1

40 } 3 5

8 } 3 (13) 1

40 } 3 5

104 } 3 1

40 } 3 5

144 } 3 5 48

The band provides 13 pounds of resistance when it is stretched to 48 inches.

5. f (x) 5 x6, x ≥ 0

1

x

y

21

y 5 x6

y 5 6x

f (x) 5 x6

y 5 x6

x 5 y6

6 6 Ï}

x 5 y

f 21(x) 5 6 Ï}

x

6. g(x) 5 1 } 27 x3

1

x

y

21

y 5 x3127

y 5 33x

y 5 1 } 27 x3

x 5 1 } 27 y3

27x 5 y3

3 Ï}

27x 5 y

3 3 Ï}

x 5 y

g21(x) 5 3 3 Ï}

x

7. f (x) 5 2 64

} 125 x3

2

x

y

21

y 5 2 x364125

y 5 2 3x5

4 y 5 2

64 } 125 x3

x 5 2 64

} 125 y3

2 125

} 64 x 5 y3

3 Î} 2 125

} 64 x 5 y

3 Ï}

2125x }

3 Ï}

64 5 y

25

3 Ï}

x }

4 5 y

f 21(x) 5 25

3 Ï}

x } 4

8. f (x) 5 2x3 1 4

1

x

y

21

y 5 2x 3 1 4

y 5 34 2x

y 5 2x3 1 4

x 5 2y3 1 4

x 2 4 5 2y3

4 2 x 5 y3

3 Ï}

4 2 x 5 y

f 21(x) 5 3 Ï}

4 2 x

9. f (x) 5 2x5 1 3

1

x

y

22

y 5 2x5 1 3

x 2 32y 5 5

y 5 2x5 1 3

x 5 2y5 1 3

x 2 3 5 2y5

1 }

2 x 2

3 } 2 5 y5

5 Î} 1 }

2 x 2

3 } 2 5 y

f 21(x) 5 5 Î} 1 }

2 x 2

3 } 2

10. g(x) 5 27x5 1 7

y 5 27x5 1 7

x 2 72 7y 5 52

x

y

2

y 5 27x5 1 7

x 5 27y5 1 7

x 2 7 5 27y5

1 2 x } 7 5 y5

5 Î} 1 2 x } 7 5 y

g21(x) 5 5 Î} 1 2 x } 7 5 5 Î}

x 2 7

} 27

11. P 5 10.7t 0.272

P }

10.7 5 t 0.272

1 P }

10.7 2

1/0.272 5 (t0.272)1/0.272

1 P }

10.7 2

3.68 5 t

When P 5 25: t 5 1 25 } 10.7 2 3.68

ø 22.7

You can predict that the average ticket price will reach $25 about 22 years after 1995, or in 2017.


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Skill Practice

1. An inverse relation interchanges the input and output values of the original relation.

2. A function g is an inverse of f provided f (g(x)) 5 x and g( f (x)) 5 x.

3. y 5 4x 2 1 4. y 5 22x 1 5

x 5 4y 2 1 x 5 22y 1 5

x 1 1 5 4y x 2 5 5 22y

1 }

4 x 1

1 } 4 5 y 2

1 } 2 x 1

5 } 2 5 y

5. y 5 7x 2 6 6. y 5 10x 2 28

x 5 7y 2 6 x 5 10y 2 28

x 1 6 5 7y x 1 28 5 10y

1 } 7 x 1

6 } 7 5 y

1 }

10 x 1

14 } 5 5 y

7. y 5 12x 1 7 8. y 5 218x 2 5

x 5 12y 1 7 x 5 218y 2 5

x 2 7 5 12y x 1 5 5 218y

1 }

12 x 2

7 } 12 5 y 2

1 } 18 x 2

5 } 18 5 y

9. y 5 5x 1 1 } 3 10. y 5 2

2 } 3 x 1 2

x 5 5y 1 1 } 3 x 5 2

2 } 3 y 1 2

x 2 1 } 3 5 5y x 2 2 5 2

2 } 3 y

1 } 5 x 2

1 } 15 5 y 2

3 } 2 x 1 3 5 y

11. y 5 2 3 } 5 x 1

7 } 5

x 5 2 3 } 5 y 1

7 } 5

x 2 7 } 5 5 2

3 } 5 y

2 5 } 3 x 1

7 } 3 5 y

12. In the last step, each term was not divided by 6.

y 5 6x 2 11

x 5 6y 2 11

x 1 11 5 6y

x }

6 1

11 } 6 5 y

13. In the fi rst step, the terms involving x and y were switched instead of just the variables x and y.

y 5 2x 1 3

x 5 2y 1 3

x 2 3 5 2y

3 2 x 5 y

14. Sample answer:

f 21(x) 5 3x 2 1

y 5 3x 2 1

x 5 3y 2 1

x 1 1 5 3y

1 }

3 x 1

1 } 3 5 y

f (x) 5 1 } 3 x 1

1 } 3

15. f (x) 5 x 1 4, g(x) 5 x 2 4

f (g(x)) 5 f (x 2 4) g( f (x)) 5 g(x 1 4)

5 x 2 4 1 4 5 x 1 4 2 4

5 x ✓ 5 x ✓

16. f (x) 5 2x 1 3, g(x) 5 1 } 2 x 2

3 } 2

f (g(x)) 5 f 1 1 } 2 x 2

3 } 2 2 g( f (x)) 5 g(2x 1 3)

5 2 1 1 } 2 x 2

3 } 2 2 1 3 5

1 } 2 (2x 1 3) 2

3 } 2

5 x 2 3 1 3 5 x 1 3 } 2 2

3 } 2

5 x ✓ 5 x ✓

17. f (x) 5 1 } 4 x3, g(x) 5 (4x)1/3

f (g(x)) 5 f [(4x)1/3] g( f (x)) 5 g 1 1 } 4 x3 2

5 1 } 4 F (4x)1/3 G 3 5 F 4 1 1 }

4 x3 2 G 1/3

5 1 } 4 (4x) 5 (x3)1/3

5 x ✓ 5 x ✓

18. f (x) 5 1 } 5 x 2 1, g(x) 5 5x 1 5

f (g(x)) 5 f (5x 1 5) g( f (x)) 5 g 1 1 } 5 x 2 1 2

5 1 } 5 (5x 1 5) 2 1 5 5 1 1 } 5 x 2 1 2 1 5

5 x 1 1 2 1 5 x 2 5 1 5

5 x ✓ 5 x ✓

19. f (x) 5 4x 1 9, g(x) 5 1 } 4 x 2

9 } 4

f (g(x)) 5 f 1 1 } 4 x 2

9 } 4 2 g( f (x)) 5 g(4x 1 9)

5 4 1 1 } 4 x 2

9 } 4 2 1 9 5

1 } 4 (4x 1 9) 2

9 } 4

5 x 2 9 1 9 5 x 1 9 } 4 2

9 } 4

5 x ✓ 5 x ✓

20. f (x) 5 5x2 2 2, x ≥ 0; g(x) 5 1 x 1 2 } 5 2 1/2

f (g(x)) 5 f F 1 x 1 2 } 5 2 1/2

G g( f (x)) 5 g(5x2 2 2)

5 5 F 1 x 1 2 } 5 2 1/2

G 2 2 2 5 1 5x2 2 2 1 2 } 5 2 1/2

5 5 1 x 1 2 } 5 2 2 2 5 1 5x2

} 5 2 1/2

5 x 1 2 2 2 5 (x2)1/2

5 x ✓ 5 x ✓


n2ws-06a.indd 378 6/27/06 10:37:05 AM

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379Algebra 2


21. B;

y 5 mx 1 b

m 5 2 } 3 , b 5 24

y 5 2 } 3 x 2 4

x 5 2 } 3 y 2 4

x 1 4 5 2 } 3 y

3 }

2 x 1 6 5 y

3 }

2 x 1 6 5 g(x)

22. f (x) 5 x7 23. f (x) 5 4x4, x ≥ 0 y 5 x7 y 5 4x4

x 5 y7 x 5 4y4

7 Ï}

x 5 y x }

4 5 y4

f 21(x) 5 7 Ï}

x 6 4 Î}

x }

4 5 y

6 4 Î}

x p 4

} 4 p 4 5 y

6 4 Ï}

4x } 2 5 y

f 21(x)5 4 Ï}

4x } 2

24. f (x) 5 210x6, x ≤ 0 y 5 210x6

x 5 210y6

2 x } 10 5 y6

6 6 Î}

2 x } 10 5 y

6 6 Î}}

2 x p 10,000

} 10 p 100,000 5 y

6 6 Ï}

2100,000x } 10 5 y

f 21(x) 5 2 6 Ï}

2100,000x } 10

25. f (x) 5 32x5

y 5 32x5

x 5 32y5

x }

32 5 y5

5 Î}

x }

32 5 y

5 Ï}

x }

5 Ï}

32 5 y

5 Ï}

x }

2 5 y

f 21(x) 5 5 Ï}

x }

2

26. f (x) 5 2 2 } 5 x3 27. f (x) 5

16 } 25 x2, x ≤ 0

y 5 2 2 } 5 x3 y 5

16 } 25 x2

x 5 2 2 } 5 y3 x 5

16 } 25 y2

2 5 } 2 x 5 y3

25 }

16 x 5 y 2

3 Î}

25x

} 2 5 y 6 Î}

25

} 16

x 5 y

3 Î} 25x p 4

} 2 p 4 5 y 6

5 } 4 Ï

}

x 5 y

3 Ï}

220x }

2 5 y 2

5 } 4 Ï

}

x 5 y

f 21(x) 5 3 Ï}

220x } 2 f 21(x) 5 2

5 } 4 Ï

}

x

28. C;

f (x) 5 2 1 } 64 x3

y 5 2 1 } 64 x3

x 5 2 1 } 64 y3

264x 5 y3

3 Ï}

264x 5 y

24 3 Ï}

x 5 y

24 3 Ï}

x 5 g(x)

29.

x

y

1

1

f (x) 5 3x 1 1

Because no horizontal line intersects the graph of f more than once, the inverse of f is a function.

30.

x

y1

1

f (x) 5 2x 2 5


31.

x

y

1

4

f (x) 5 x2 2 114

Because a horizontal line intersects the graph of f more than once, the inverse of f is not a function.


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32.

x

y1

1

f (x) 5 26x2


33.

x

y

1

1

f (x) 5 3x3


34.

x

y

1

21

f (x) 5 x3 2 2


35.

x

y

2

2

f(x) 5 (x 2 4)(x 1 1)


36.

x

y

2

2

f(x) 5 ux u 1 4


37.

x

y2

2

f(x) 5 4x4 2 5x2 2 6


38. f (x) 5 3 } 2 x4, x ≥ 0 39. f (x) 5 x3 2 2

y 5 3 } 2 x4 y 5 x3 2 2

x 5 3 } 2 y4 x 5 y 3 2 2

2 }

3 x 5 y4 x 1 2 5 y3

6 4 Î}

2x

} 3 5 y

3 Ï}

x 1 2 5 y

6 4 Î} 2x p 27

} 3 p 27

5 y f 21(x) 5 3 Ï}

x 1 2

6 4 Ï}

54x } 3 5 y

f 21(x) 5 4 Ï}

54x } 3

40. f (x) 5 3 } 4 x5 1 5

y 5 3 } 4 x5 1 5

x 5 3 } 4 y5 1 5

x 2 5 5 3 } 4 y5

4 }

3 x 2

20 } 3 5 y5

5 Î} 4 }

3 x 2

20 } 3 5 y

f 21(x) 5 5 Î} 4 }

3 x 2

20 } 3

41. f (x) 5 2 2 } 5 x6 1 8, x ≤ 0

y 5 2 2 } 5 x6 1 8

x 5 2 2 } 5 y6 1 8

x 2 8 5 2 2 } 5 y6

2 5 } 2 x 1 20 5 y6

6 6 Î} 2

5 } 2 x 1 20 5 y

f 21(x) 5 2 6 Î} 2

5 } 2 x 1 20

42. f (x) 5 2x3 2 6

} 9

y 5 2x3 2 6

} 9

x 5 2y3 2 6

} 9

9x 5 2y3 2 6

9x 1 6 5 2y3

9 }

2 x 1 3 5 y3

3 Î} 9 }

2 x 1 3 5 y

f 21(x) 5 3 Î} 9 }

2 x 1 3


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381Algebra 2


43. f (x) 5 x4 2 9, x ≥ 0 y 5 x4 2 9

x 5 y4 2 9

x 1 9 5 y4

6 4 Ï}

x 1 9 5 y

f 21(x) 5 4 Ï}

x 1 9 44. a. False

x

y

1

1f(x) 5 x2


b. True

x

y

1

1

f(x) 5 x3


45. f (x) 5 mx 1 b

y 5 mx 1 b

x 5 my 1 b

x 2 b 5 my

x 2 b

} m

5 y

1 }

m x 2

b } m 5 y

f 21(x) 5 1 } m x 2

b } m

Slope: 1 }

m y-intercept: 2

b } m

This is the graph of a line, and it is a function if m Þ 0.

Problem Solving

46. E 5 0.81419D

E }

0.81419 5 D

When E 5 250: D 5 E } 0.81419 5

250 } 0.81419 ø 307

The amount that could be obtained for 250 euros was about $307.

47. a. l 5 0.5w 1 3

l 2 3 5 0.5w

l 2 3

} 0.5

5 w

2l 2 6 5 w

b. When l 5 6.5: w 5 2(6.5) 2 6 5 7

The weight is 7 pounds.

48. a. C 5 5 } 9 (F 2 32)

9 } 5 C 5 F 2 32

9 } 5 C 1 32 5 F

From the inverse, you can convert temperatures from degrees Celsius C to degrees Fahrenheit F.

b. When C 5 5: F 5 9 } 5 (5) 1 32 5 9 1 32 5 41

At the start of the race, the temperature was 418F.

When C 5 210: F 5 9 } 5 (210) 1 32 5 218 1 32 5 14

At the end of the race, the temperature was 148F.

c.

IntersectionX=-40 Y=-40

The temperature that is the same on both temperature scales is 240.

49. v 5 1.34 Ï}

l�

v }

1.34 5 Ï

}

l�

1 v }

1.34 2

2 5 l

When v 5 7.5: l 5 1 7.5 } 1.34 2 2 ø 31.3

A water line length of about 31.3 feet is needed to achieve a maximum speed of 7.5 knots.

50. A 5 0.2195h0.3964

A }

0.2195 5 h0.3964

1 A }

0.2195 2 1/0.3964

5 (h0.3964)1/0.3964

1 A }

0.2195 2 2.523

5 h

When A 5 1.6: h 5 1 1.6 } 0.2195 2 2.523

ø (7.29)2.523 ø 150

The height of a 60 kilogram person who has a body surface area of 1.6 square meters is about 150 centimeters.

51. a.

x

y

1

1

g(x) 5 2x

The function g(x) 5 2x is its own inverse because the graph of an inverse relation is a refl ection of the graph of the original relation in the line y 5 x and the graph of an inverse relation is the same as the graph of the original relation.

g(x) 5 2x

y 5 2x

x 5 2y

2x 5 y

2x 5 g21(x)


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b.

x

y

4

4

g(x) 5 2x 2 5

g(x) 5 2x 1 7

g(x) 5 2x 1 12

c. g(x) 5 2x 1 b, where b is any real number.

52.

1

x

y

22

53.

1

x

y

21

54. 1

x

y

21

55. 3

x

y

21

56. 1

x

y

21

57.

1

x

y

21

58. 3x 2 4y 5 24

x 1 2y 5 2 → x 5 22y 1 2

3(22y 1 2) 2 4y 5 24 x 5 22 1 2 9 } 5 2 1 2

26y 1 6 2 4y 5 24 x 5 18

} 5 1 2

210y 5 18 x 5 28

} 5

y 5 2 9 } 5

The solution is 1 28 } 5 , 2

9 } 5 2 .

59. 2x 2 4y 5 13 3 (22) 24x 1 8y 5 226

4x 2 5y 5 8 4x 2 5y 5 8

3y 5 218

y 5 26

2x 2 4(26) 5 13

2x 1 24 5 13

2x 5 211

x 5 2 11

} 2

The solution is 1 2 11

} 2 , 26 2 .

60. 7x 2 12y 5 222 3 2 14x 2 24y 5 244

25x 1 8y 5 14 3 3 75x 1 24y 5 42

89x 5 22

x 5 22

}89

7 12 2}89 2 2 12y 5 222

214}89 2 12y 5 222

212y 5 21944}

89

y 5 162}89

The solution is 12 2}89 ,

162}89 2 .

61. 6x3 2 54x 5 0

6x(x2 2 9) 5 0

6x(x 1 3)(x 2 3) 5 0

x 5 0, x 5 23, or x 5 3

62. 2x3 2 8x2 1 8x 5 0

2x(x2 2 4x 1 4) 5 0

2x(x 2 2)2 5 0

x 5 0 or x 5 2

63. 16x3 5 2250

16x3 1 250 5 0

2(8x3 1 125) 5 0

2[(2x)3 1 53] 5 0

2[(2x 1 5)(4x2 2 10x 1 25)] 5 0

x 5 2 5 } 2 or x 5

2(210) 6 Ï}}

(210)2 2 4(4)(25) }}}

2(4)

5 10 6 Ï

}

2300 } 8 5

10 6 10 Ï}

3 i } 8 5

5 6 5 Ï}

3 i } 4

64. x3 2 3x2 1 8x 5 24

x3 2 3x2 1 8x 2 24 5 0

x2(x 2 3) 1 8(x 2 3) 5 0

(x 2 3)(x2 1 8) 5 0

x 5 3 or x2 1 8 5 0

x2 5 28

x 5 6 Ï}

28 5 62 Ï}

2 i

x 5 2 Ï}

2 i, x 5 22 Ï}

2 i, or x 5 3

65. 4x3 1 8x2 2 25x 2 50 5 0

4x2(x 1 2) 2 25(x 1 2) 5 0

(4x2 2 25)(x 1 2) 5 0

(2x 1 5)(2x 2 5)(x 1 2) 5 0

x 5 25}2 , x 5

5}2 , or x 5 22


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383Algebra 2


66. 12x4 2 7x2 2 45 5 0

(4x2 2 9)(3x2 1 5) 5 0

(2x 1 3)(2x 2 3)(3x2 1 5) 5 0

x 5 6 3 } 2 or 3x2 1 5 5 0

3x2 5 25

x2 5 2 5 } 3

x 5 6 Ï}

25

} 3 5 6

Ï}

25 }

Ï}

3 p Ï

}

3 }

Ï}

3 5 6

Ï}

215 } 3 5 6

i Ï}

15 } 3

x 5 2 3 } 2 , x 5

3 } 2 , x 5

i Ï}

15 } 3 , or x 5

2i Ï}

15 } 3

Quiz 6.3–6.4 (p. 445)

1. f (x) 1 g (x) 5 4x2 2 x 1 2x2 5 6x2 2 x

The function f and g each have the same domain: All real numbers. So, the domain of f 1 g also consists of all real numbers.

2. g (x) 2 f (x) 5 2x2 2 (4x2 2 x) 5 2x2 2 4x2 1 x 5 22x2 1 x

The function f and g each have the same domain: All real numbers. So, the domain of f 2 g also consists of all real numbers.

3. f (x) p g (x) 5 (4x2 2 x)(2x2) 5 8x4 2 2x3

The function f and g each have the same domain: All real numbers. So, the domain of f p g also consists of all real numbers.

4. f (x)

} g (x)

5 4x2 2 x

} 2x2 5

4x2

} 2x2 2

x }

2x2 5 2 2 1 } 2x

The function f and g each have the same domain: All real

numbers. Because g(0) 5 0, the domain of f } g is restricted

to all real numbers except x 5 0.

5. f (g(x)) 5 f (2x2) 5 4(2x2)2 2 2x2 5 4(4x4) 2 2x2

5 16x4 2 2x2

The domain of f (g (x)) consists of all real numbers.

6. g ( f (x)) 5 g(4x2 2 x) 5 2(4x2 2 x)2

5 2(4x2 2 x)(4x2 2 x)

5 2(16x4 2 8x3 1 x2) 5 32x4 2 16x3 1 2x2

The domain of g ( f (x)) consist of all real numbers.

7. f ( f (x)) 5 f (4x2 2 x) 5 4(4x2 2 x)2 2 (4x2 2 x) 5 4(4x2 2 x)(4x2 2 x) 2 4x2 1 x

5 4(16x4 2 8x3 1 x2) 2 4x2 1 x

5 64x4 2 32x3 1 4x2 2 4x2 1 x

5 64x4 2 32x3 1 x

The domain of f ( f (x)) consists of all real numbers.

8. g ( g(x)) 5 g(2x2) 5 2(2x2)2 5 2(4x4) 5 8x4

The domain of g (g (x)) consists of all real numbers.

9. f (x) 5 x 2 9, g (x) 5 x 1 9

f ( g (x)) 5 f (x 1 9) g ( f (x)) 5 g (x 2 9)

5 x 1 9 2 9 5 x 2 9 1 9

5 x ✓ 5 x ✓

10. f (x) 5 5x3, g (x) 5 3 Î}

x } 5

f ( g (x)) 5 f 1 3 Î}

x } 5 2 g (f ( x)) 5 g (5x3)

5 5 1 3 Î}

x } 5 2 3 5 3

Î}

5x3

} 5

5 5 1 x } 5 2 5 3 Î}

x3

5 x ✓ 5 x ✓

11. f (x) 5 2 3 } 2 x 1

1 } 4 , g (x) 5 2

2 } 3 x1

1 } 6

f (g (x)) 5 f 1 2 2 } 3 x 1

1 } 6 2 g (f (x)) 5 g 1 2

3 } 2 x 1

1 } 4 2

5 2 3 } 2 1 2

2 } 3 x 1

1 } 6 2 1

1 } 4 5 2

2 } 3 1 2

3 } 2 x 1

1 } 4 2 1

1 } 6

5 x 2 1 } 4 1

1 } 4 5 x 2

1 } 6 1

1 } 6

5 x ✓ 5 x ✓

12. f (x) 5 6x2 1 1, x ≥ 0; g (x) 5 1 x 2 1 } 6 2 1/2

f (g (x)) 5 fF 1 x 2 1 }

6 2 1/2G g ( f (x)) 5 g(6x2 1 1)

5 6F 1 x 2 1 }

6 2 1/2G2

1 1 5 1 6x2 1 1 2 1 } 6 2

1/2

5 6 1 x 2 1 }

6 2 1 1 5 1 6x2

} 6 2 1/2

5 x 2 1 1 1 5 (x2)1/2

5 x ✓ 5 x ✓

13. f (x) 5 2 1 } 3 x 1 5

y 5 2 1 } 3 x 1 5

x 5 2 1 } 3 y 1 5

x 2 5 5 2 1 } 3 y

23x 1 15 5 y

f 21(x) 5 23x 1 15

14. f ( x) 5 x2 2 16, x ≥ 0 y 5 x2 2 16

x 5 y2 2 16

x 1 16 5 y2

6 Ï}

x 1 16 5 y

f 21(x) 5 Ï}

x 1 16


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15. f (x) 5 2 2 } 9 x5

y 5 2 2 } 9 x5

x 5 2 2 } 9 y5

2 9 } 2 x 5 y5

5 Î}

2 9x

} 2 5 y

5 Î} 2

9x p 16 } 2 p 16 5 y

5 Ï}

2144x }

2 5 y

f 21(x) 5 5 Ï}

2144x } 2

16. f (x) 5 5x 1 12

y 5 5x 1 12

x 5 5y 1 12

x 2 12 5 5y

1 } 5 x 2

12 } 5 5 y

f 21(x) 5 1 } 5 x 2

12 } 5

17. f (x) 5 23x3 2 4

y 5 23x3 2 4

x 5 23y3 2 4

x 1 4 5 23y3

x 1 4

} 23

5 y3

3 Î}

x 1 4

} 23

5 y

3 Î} (x 1 4)9

} 23 p 9 5 y

2 3 Ï}

9x 1 36 } 3 5 y

f 21(x) 5 2 3 Ï}

9x 1 36 } 3

18. f (x) 5 9x4 2 49, x ≤ 0

y 5 9x4 2 49

x 5 9y4 2 49

x 1 49 5 9y4

x 1 49

} 9 5 y4

6 4 Î} x 1 49

} 9 5 y

6 4 Ï}

(x 1 49)9

} 9 p 9 5 y

6 4 Ï}

9x 1 441 } 3 5 y

f 21(x) 5 2 4 Ï}

9x 1 441 } 3

19. C (g (d)) 5 C(0.02d) 5 2.15(0.02d) 5 0.043d

C (g (400)) 5 0.043(400) 5 17.2

The expression C(g(400)) or $17.20 represents the cost of gasoline for the car that is driven 400 miles.

Lesson 6.5


1. y 5 23 Ï}

x

1

x

y

21

x 0 1 2 3 4

y 0 23 24.24 25.2 26

The domain is x ≥ 0 and the range is y ≤ 0.

2. f (x) 5 1 } 4 Ï

}

x

1

x

y

21

x 0 1 2 3 4

f (x) 0 0.25 0.35 0.43 0.5

The domain is x ≥ 0 and range is y ≥ 0.

3. y 5 2 1 } 2 3 Ï}

x

1x

y

22

x 22 21 0 1 2

f (x) 20.63 0.5 0 20.5 20.63

The domain and range are all real numbers.


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385Algebra 2


4. g (x) 5 4 3 Ï}

x

2

x

y

22

x 22 21 0 1 2

f (x) 25.04 24 0 4 5.04


5.

X=.80851064 Y=.99808114

A pendulum with a period of 1 second is about 0.81 foot long.

6. y 5 24 Ï}

x 1 2

1

x

y

21

Because h 5 0 and k 5 2, shift the graph of y 5 24 Ï}

x up 2 units.


7. y 5 2 Ï}

x 1 1

1

x

y

21


x left 1 unit. The domain is x ≥ 21and the range is y ≥ 0.

8. f (x) 5 1 } 2 Ï}

x 2 3 2 1

1

x

y

21

Because h 5 3 and k 5 21, shift the graph of

f (x) 5 1 } 2 Ï

}

x right 3 units and down 1 unit.

The domain is x ≥ 3 and the range is f (x) ≥ 21.

9. y 5 2 3 Ï}

x 2 4

1

x

y

21

Because h = 4 and k 5 0, shift the graph of y 5 2 3 Ï}

x right 4 units.

The domain and the range are both all real numbers.

10. y 5 3 Î}

x 2 5

1

x

y

21

Because h 5 0 and k 5 25, shift the graph of y 3 Ï}

x down 5 units.

The domain and range are both all real numbers.

11. g (x) 5 2 3 Î}

x 1 2 2 3

1

x

y

21

Because h 5 22 and k 5 23, shift the graph of g (x) 5 2

3 Ï}

x left 2 units and down 3 units.



Skill Practice

1. Square root functions and cube root functions are examples of radical functions.

2. a. When a 5 23, the graph of y 5 Ï}

x is stretched vertically by a factor of 3 and then refl ected in the x-axis.

b. When h 5 2, the graph of y 5 Ï}

x is shifted to the right 2 units.

c. When k 5 4, the graph of y 5 Ï}

x is shifted up 4 units.


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3. y 5 24 Ï}

x

x 0 1 2 3 4

y 0 24 25.66 26.93 28

21x

y

1


4. f (x) 5 1 } 2 Ï

}

x

x 0 1 2 3 4

f (x) 0 0.5 0.71 0.87 1

1

x

y

21


5. y 5 2 4 } 5 Ï

}

x

x 0 1 2 3 4

y 0 20.8 21.13 21.39 21.6

1

x

y

21


6. y 5 26 Ï}

x

x 0 1 2 3 4

y 0 26 28.49 210.39 212

2

x

y

21


7. y 5 5 Ï}

x

x 0 1 2 3 4

y 0 5 7.07 8.66 10

2

x

y

21

The domain is x ≥ 0 and the range is y ≥ 0.

8. g (x) 5 9 Ï}

x

x 0 1 2 3 4

g(x) 0 9 12.73 15.59 18

3

x

y

21

The domain is x ≥ and the range is g (x) ≥ 0.

9. D;

y 5 2 3 } 2 Ï

}

x

x 0 1 2 3 4

y 0 21.5 22.12 22.6 23

10. y 5 1 } 4

3 Î}

x

x 22 21 0 1 2

y 20.32 20.25 0 0.25 0.31

x

y

22

14



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387Algebra 2


11. y 5 2 3 Î}

x

x 22 21 0 1 2

y 22.52 22 0 2 2.52

1

x

y

22


12. f (x) 5 25 3 Î}

x

x 22 21 0 1 2

f (x) 6.3 5 0 25 26.3

2

x

y

22


13. h (x) 5 2 1 } 7 3

Î}

x

x 22 21 0 1 2

h (x) 0.18 0.14 0 20.14 20.18

.2

x

y

22


14. g (x) 5 6 3 Î}

x

x 22 21 0 1 2

g(x) 27.56 26 0 6 7.56

3

x

y

22


15. y 5 7 } 9

3 Î}

x

x 22 21 0 1 2

h (x) 20.98 20.78 0 0.78 0.98

1

x

y

22


16. f (x) 5 2 Ï}

x 2 1 1 3

1

x

y

21

Because h 5 1 and k 5 3, shift the graph of f (x) 5 2 Ï}

x right 1 unit and up 3 units.


17. y 5 (x 1 1)1/2 1 8 5 Ï}

x 1 1 1 8

2

x

y

21

Because h 5 21 and k 5 8, shift the graph of y 5 x1/2 left 1 unit and up 8 units.



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18. y 5 24 Ï}

x 2 5 1 1

1

x

y

21


x right 5 units and up 1 unit.


19. y 5 3 } 4 x1/321 5

3 } 4

3 Î}

x 21

1

x

y

22


y 5 3 } 4 x1/3 down 1 unit.


20. y 5 22 3 Î}

x 1 5 1 5

1

x

y

22


y 5 22 3 Ï}

x left 5 units and up 5 units.


21. h (x) 5 23 3 Î}

x 1 7 2 6

2

x

y

22

Because h 5 27 and k 5 26, shift the graph of h (x) 5 23

3 Ï}



22. y 5 2 Ï}

x 2 4 2 7

2

x

y

22


x right 4 units and down 7 units.


23. g (x) 5 2 1 } 3 3 Ï}

x 2 6

1

x

y

22

Because h 5 0 and k 5 26 shift the graph of

g (x) 5 1 } 3 3 Ï}

x down 6 units.


24. y 5 4 3 Î}

x 2 4 1 5

2

x

y

21

Because h 5 4 and k 5 5, shift the graph of y 5 4 3 Ï}

x right 4 units and up 5 units.


25. The domain of y 5 Ï}

x 2 5 1 4 is limited because of the square root of a negative number is not a real number. Because the domain is restricted, the range is also affected.

26. The error occured in describing the horizontal translation.

Because h 5 21, the graph is translated left 1 unit. So,

graph of y 5 22 3 Î}

x 1 1 2 3 is the graph of

y 5 22 3 Î}

x translated left 1 unit and down 3 units.

27. C;

Because the graph of y 5 3 3 Î}

x is shifted left 2 units, h 5 22 and k 5 0. An equation of the translated graph is y 5 3

3 Î}

x 1 2 .

28. The function y 5 Ï}

x 1 5 is a translation of the function y 5 Ï

}

x that has a domain of x ≥ 0 and a range of y ≥ 0. Because h 5 25, the domain is x ≥ 25. Because k 5 0, the range is y ≥ 0.


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389Algebra 2


29. The function y 5 Ï}

x 2 12 is a translation of the function y 5 Ï

}

x that has a domain x ≥ 0 and a range of y ≥ 0. Because h 5 12, the domain is x ≥ 12. Because k 5 0, the range is y ≥ 0.

30. The function y 5 1 } 3 Ï

}

x 2 4 is a translation of the function

y 5 1 } 3 Ï

}

x that has a domain of x ≥ 0 and a range of y ≥ 0.

Because h 5 0, the domain is x ≥ 0. Because k 5 24, the

range is y ≥ 24.

31. The function y 5 1 } 2

3 Î}

x 1 7 is a translation of the function

y 5 1 } 2 3 Ï}

x that has a domain and range of all real numbers.

Therefore, the function y 5 1 } 2

3 Î}

x 1 7 also has a domain

and range of all real numbers.

32. The function g (x) 5 3 Î}

x 1 7 is a translation of the function g (x) 5

3 Î}

x that has a domain of all real numbers. Therefore, the function g (x) 5 3

Î}

x 1 7 also has a domain and range of all real numbers.

33. The function f (x) 5 1 } 4 Ï}

x 2 3 1 6 is a translation of the

function f (x) 5 1 } 4 Ï

}

x that has a domain of x ≥ 0 and a range

of f (x) ≥ 0. Because h 5 3, the domain is x ≥ 3. Because k 5 6, the range is f (x) ≥ 6.

34.

When n is even, the domain of the function y 5 n Ï}

x is x ≥ 0 and the range of the function is y ≥ 0. When n is odd, the domain and the range of the function y 5

n Ï}

x are both all real numbers.

Problem Solving

35.

X=43.03 Y=8

You can see 8 miles at an altitude of about 43 feet above sea level.

36. a.

X=2.01 Y=1.57

A pendulum with a length of 2 feet has a period of about 1.6 seconds.

b.

X=3.26 Y=2

A pendulum with a period of 2 seconds is about 3.3 feet long.

37. a. v 5 331.5 Î} K }

273.15 , K ≥ 0

5 331.5 Î} 273.15 1 C

} 273.15

5 331.5 Î} 1 1 C } 273.15

b. Because K ≥ 0 and K 5 273.15 1 C:

273.15 1 C ≥ 0 C ≥ 2273.15

The domain of the function is C ≥ 2273.15.

The range of the function is v ≥ 0.

38.

0 1000 2000 30000

50

100

150

200

250

Power (horsepower)

Sp

eed

(m

ph

)

p

s

The power of a 3500 pound car that reaches a speed of 200 miles per hour is about 2500 horsepower.

39. a. vt 5 33.7 Î}

W

} A

5 33.7 Î}

165

} A

b. A 2 4 6 8 10

vt 306.1 216.44 176.72 153.05 136.89

c.

0 2 4 6 8 10 A

Vt

0

100

200

300

400

500


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40. a. s 5 πr 1 πr2

s 5 π(r 1 r2)

s 1 π 1 1 } 4 2 5 π 1 r2 1 r 1 1 } 4 2

s 1 π

} 4 5 π 1 r 1 1 } 2 2 2

1 } π� 1 s 1

π } 4 2 5 1 r 1

1 } 2 2 2

Ï}

1 } π� 1 s 1

π } 4 2 5 r 1

1 } 2

1 }

Ï}

π Ï}

1 s 1 π

} 4 2 5 r 1 1 } 2

1 }

Ï}

π Ï}

s 1 π

} 4 2 1 } 2 5 r

b.

c. r 5 1 }

Ï}

π� Ï}

s 1 π

} 4 2 1 } 2

r 5 1 }

Ï}

π� Ï}

3 π

} 4 1

π } 4 2

1 } 2

r 5 1 }

Ï}

π� p Ï

}

π� 2 1 } 2

r 5 0.5

The radius is 0.5 unit.

Mixed Review

41. 1 } 5 (x 1 8)2 5 3

(x 1 8)2 5 15

x 1 8 5 6 Ï}

15

x 5 28 6 Ï}

15

42. 9(x 2 3)2 1 22 5 130

9(x 2 3)2 5 108

(x 2 3)2 5 12

x 2 3 5 6 Ï}

12

x 5 3 6 2 Ï}

3

43. 7x2 2 20 5 36

7x2 5 56

x2 5 8

x 5 6 Ï}

8

x 5 6 2 Ï}

2

44. x2 2 14x 1 37 5 0

x2 2 14x 5 237

x2 2 14x 1 49 5 237 1 49

(x 2 7)2 5 12

x 2 7 5 6 Ï}

12

x 5 7 6 2 Ï}

3

45. x2 2 40x 1 383 5 0

x2 2 40x 5 2383

x2 2 40x 1 400 5 2383 1 400

(x 2 20)2 5 17

x 2 20 5 6 Ï}

17

x 5 20 6 Ï}

17

46. x2 2 22x 1 97 5 0

x2 2 22x 5 297

x2 2 22x 1 121 5 297 1 121

(x 2 11)2 5 24

x 2 11 5 6 Ï}

24

x 5 11 6 2 Ï}

6

47. f (x) 5 x4 1 5x3 2 3x2 2 9x

5 x(x3 1 5x2 2 3x 2 9) Using factoring and the zero (or root) feature of a graphing

calculator, the zeros are x 5 0, x ø 25.2, x ø 21.2, and x ø 1.4.

48. f (x) 5 x4 2 7x3 2 4x2 2 8x 1 22

The possible rational zeros are 61, 62, 611, and 622. Because none of these satisfy f (x) 5 0, there are no rational zeros.

Using the zero (or root) feature of a graphing calculator, the real zeros are x ø 1.1 and x ø 7.6. The two remaining zeros are complex conjugates. One factor of the polynomial can be approximated as (x 2 7.6)(x 2 1.1) 5 x2 2 8.7x 1 8.36.

Using long division:

x4 2 7x3 2 4x2 2 8x 1 22

}} x2 2 8.7x 1 8.36

ø x2 1 1.7x 1 2.43

So, x ø 21.7 6 Ï

}}

1.72 2 4(1)(2.43) }}}

2(1)

5 20.85 6 Ï}

26.83 } 2 5 20.85 6 1.3i

So, the zeros of the function x ø 1.1, x ø 7.6, andx ø 20.85 6 1.3i.


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391Algebra 2


49. f (x) 5 x4 1 x3 1 2x2 1 4x 2 8

Possible rational zeros: 61, 62, 64, 68

1 1 1 2 4 28

1 2 4 8

1 2 4 8 0

f (x) 5 (x 2 1)(x3 1 2x2 1 4x 1 8)

5 (x 2 1) F x2(x 1 2) 1 4(x 1 2) G 5 (x 2 1)(x 1 2)(x2 1 4) 5 (x 2 1)(x 1 2)(x 1 2i)(x 2 2i)

The zeros of the function are 1, 22, 22i, and 2i.

50. f (x) 5 x4 1 6x3 1 14x2 1 54x 1 45

Possible rational zeros: 61, 63, 65, 69, 615, 645

21 1 6 14 54 45

21 25 29 245

1 5 9 45 0

f (x) 5 (x 1 1)(x3 1 5x2 1 9x 1 45)

5 (x 1 1) F x2(x 1 5) 1 9(x 1 5) G 5 (x 1 1)(x 1 5)(x2 1 9)

5 (x 1 1)(x 1 5)(x 1 3i)(x 2 3i)

The zeros of the function are 21, 25, 23i, and 3i.

51. f (x)

} g (x)

5 5x2/3

} 23x1/2 5 2

5x(2/3 2 1/2)

} 3 5 2 5x1/6

} 3

52. f (x) p f (x) 5 5x2/3 p 5x2/3 5 25x(2/3 1 2/3) 5 25x4/3

53. g (x) p g (x) 5 23x1/2 p (23x1/2) 5 9x(1/2 1 1/2) 5 9x

54. f (x) p g(x) 5 5x2/3 p (23x1/2) 5 215x(2/3 1 1/2) 5 215x7/6

55. f (g (x)) 5 f (23x1/2) 5 5(23x1/2)2/3

5 5[(23)2/3 p (x1/2)2/3] 5 5((23)2)1/3 p x1/3 5 5(91/3 p x1/3) 5 5

3 Î}

9x

56. g ( f (x)) 5 g(5x2/3) 5 23(5x2/3)1/2 5 23 p 51/2x1/3

57. f ( f (x)) 5 f (5x2/3) 5 5(5x2/3)2/3 5 55/3x4/9

58. g (g (x)) 5 0, x 5 0

g (g (x)) is defi ned only at x 5 0.

Lesson 6.6


1. 3 Î}

x 2 9 5 21

3 Î}

x 5 8

( 3 Î} x )3 5 83

x 5 512

Check: 3 Î}

512 2 9 0 21

8 2 9 0 21

21 5 21 ✓

2. Ï}

x 1 25 5 4

( Ï}

x 1 25 )2 5 42

x 1 25 5 16

x 5 29

Check: Ï}

29 1 25 0 4

Ï}

16 0 4

4 5 4 ✓

3. 2 3 Î}

x 2 3 5 4

3 Î}

x 2 3 5 2

( 3 Ï}

x 2 3 )3 5 23

x 2 3 5 8

x 5 11

Check: 2 3 Î}

11 2 3 0 4

2 3 Î}

8 0 4

4 5 4 ✓

4. v (p) 5 6.3 Ï}

1013 2 p

48.3 5 6.3 Ï}

1013 2 p

7.67 ø Ï}

1013 2 p

(7.67)2 ø ( Ï}

1013 2 p )2

58.8 ø 1013 2 p

2954.2 ø 2p

954.2 ø p The air pressure at the center of the hurricane is about

954 millibars.

5. 3x3/2 5 375

x3/2 5 125

(x3/2)2/3 5 1252/3

x 5 25

Check: 3(25)3/2 0 375

3(125) 0 375

375 5 375 ✓

6. 22x3/4 5 216

x3/4 5 8

(x3/4)4/3 5 84/3

x 5 16

Check: 22(16)3/4 0 216

22(8) 0 216

216 5 216 ✓

7. 2 2 } 3 x1/5 5 22

x1/5 5 3

(x1/5)5 5 35

x 5 243

Check:2 2 } 3 (243)1/5 0 22

2 2 } 3 (3) 0 22

22 5 22 ✓


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8. (x 1 3)5/2 5 32

[(x 1 3)5/2]2/5 5 322/5

x 1 3 5 4

x 5 1

Check: (1 1 3)5/2 0 32

45/2 0 32

32 5 32 ✓

9. (x 2 5)4/3 5 81

[(x 2 5)4/3]3/4 5 813/4

x 2 5 5 6(811/4)3

x 2 5 5 6(3)3

x 2 5 5 627

x 5 32 or x 5 222

Check x 5 32: Check x 5 222:

(32 2 5)4/3 0 81 (222 2 5)4/3 0 81

274/3 0 81 (227)4/3 0 81

81 5 81 ✓ (23)4 0 81

81 5 81 ✓

10. (x 1 2)2/3 1 3 5 7

(x 1 2)2/3 5 4

[(x 1 2)2/3]3/2 5 43/2

x 1 2 5 (41/2)3

x 1 2 5 (62)3

x 1 2 5 68

x 5 6 or x 5 210


(6 1 2)2/3 1 3 0 7 (210 1 2)2/3 1 3 0 7

82/3 1 3 0 7 (28)2/3 1 3 0 7

4 1 3 0 7 4 1 3 0 7

7 5 7 ✓ 7 5 7 ✓

11. x 2 1 } 2 5 Î}

1 }

4 x

1 x 2 1 } 2 2 2 5 1 Ï

}

1 }

4 x 2

2

x2 2 x 1 1 } 4 5

1 } 4 x

x2 2 5 } 4 x 1

1 } 4 5 0

(x 2 1) 1 x 2 1 } 4 2 5 0

x 5 1 or x 5 1 } 4

Check x 5 1: Check x 5 1 } 4 :

1 2 1 } 2 0 Ï

}

1 }

4 (1)

1 }

4 2

1 } 2 0 Ï

}

1 }

4 1 1 } 4 2

1 }

2 0 Ï

}

1 }

4 2

1 } 4 0 Ï

}

1 }

16

1 }

2 5

1 } 2 ✓ 2

1 } 4 Þ

1 } 4

The only solution is 1.

12. Ï}

10x 1 9 5 x 1 3

( Ï}

10x 1 9 )2 5 (x 1 3)2

10x 1 9 5 x2 1 6x 1 9

0 5 x2 2 4x

0 5 x(x 2 4)

x 5 0 or x 5 4


Ï}

10(0) 1 9 0 0 1 3 Ï}

10(4) 1 9 0 4 1 3

Ï}

9 0 3 Ï}

49 0 7

3 5 3 ✓ 7 5 7 ✓

The solutions are 0 and 4.

13. Ï}

2x 1 5 5 Ï}

x 1 7

( Ï}

2x 1 5 )2 5 ( Ï

} x 1 7 )2

2x 1 5 5 x 1 7

x 5 2

Check: Ï}

2(2) 1 5 0 Ï}

2 1 7

3 5 3 ✓

14. Ï}

x 1 6 2 2 5 Ï}

x 2 2

( Ï}

x 1 6 2 2)2 5 ( Ï

}

x 2 2 )2

x 1 6 2 4 Ï}

x 1 6 1 4 5 x 2 2

24 Ï}

x 1 6 5 212

Ï}

x 1 6 5 3

( Ï}

x 1 6 )2 5 32

x 1 6 5 9

x 5 3

Check: Ï}

3 1 6 2 2 0 Ï}

3 2 2

Ï}

9 2 2 0 Ï}

1

3 2 2 0 1

1 5 1 ✓


Skill Practice

1. When you solve an equation algebraically, an apparent solution that must be rejected because it does not satisfy the original equation is called an extraneous solution.

2. To avoid this solution, the student could have added Ï}

9x 2 5 to each side of the equation before squaring each side.

3. Ï}

5x 1 1 5 6

( Ï}

5x 1 1 )2 5 62

5x 1 1 5 36

5x 5 35

x 5 7

Check: Ï}

5(7) 1 1 0 6

Ï}

36 0 6

6 5 6 ✓


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393Algebra 2


4. Ï}

3x 1 10 5 8

( Ï}

3x 1 10 )2 5 82

3x 1 10 5 64

3x 5 54

x 5 18

Check: Ï}

3(18) 1 10 0 8

Ï}

64 0 8

8 5 8 ✓

5. Ï}

9x 1 11 5 14

Ï}

9x 5 3

( Ï}

9x )2 5 32

9x 5 9

x 5 1

Check: Ï}

9(1) 1 11 0 14

Ï}

9 1 11 0 14

14 5 14 ✓

6. Ï}

2x 2 2 } 3 5 0

Ï}

2x 5 2 } 3

( Ï}

2x )2 5 1 2 }

3 2

2

2x 5 4 } 9

x 5 2 } 9

Check: Ï}

2 1 2 } 9 2 2

2 } 3 0 0

Ï}

4 }

9 2

2 } 3 0 0

0 5 0 ✓

7. 22 Ï}

24x 1 13 5 211

22 Ï}

24x 5 224

Ï}

24x 5 12

( Ï}

24x )2 5 122

24x 5 144

x 5 6

Check: 22 Ï}

24(6) 1 13 0 211

22 Ï}

144 1 13 0 211

22(12) 1 13 0 211

211 5 211 ✓

8. 8 Ï}

10x 2 7 5 9

8 Ï}

10x 5 16

Ï}

10x 5 2

( Ï}

10x )2 5 22

10x 5 4

x 5 2 } 5

Check: 8 Ï}

10 1 2 } 5 2 2 7 0 9

8(2) 2 7 0 9

9 5 9 ✓

9. Ï}

x 2 25 1 3 5 5

Ï}

x 2 25 5 2

( Ï}

x 2 25 )2 5 22

x 2 25 5 4

x 5 29

Check: Ï}

29 2 25 1 3 0 5

2 1 3 0 5

5 5 5 ✓

10. 24 Ï}

x 2 6 5 220

24 Ï}

x 5 214

Ï}

x 5 7 } 2

( Ï}

x )2 5 1 7 } 2 2 2

x 5 49

} 4

Check: 24 Ï}

49

} 4 2 6 0 220

24 1 7 } 2 2 2 6 0 220

214 2 6 0 220

220 5 220

11. Ï}

22x 1 3 2 2 5 10

Ï}

22x 1 3 5 12

( Ï}

22x 1 3 )2 5 122

22x 1 3 5 144

22x 5 141

x 5 2 141

} 2

Check: Ï}}

22 1 2 141

} 2 2 1 3 2 2 0 10

Ï}

144 2 2 0 10

10 5 10 ✓

12. C;

Ï}

8x 1 3 5 3

( Ï}

8x 1 3 )2 5 32

8x 1 3 5 9

8x 5 6

x 5 3 } 4

Check: Ï}

8 1 3 } 4 2 1 3 0 3

Ï}

6 1 3 0 9

Ï}

9 0 3

3 5 3 ✓


n2ws-06b.indd 393 6/27/06 10:40:17 AM

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13. 3 Ï

}

x 2 10 5 23

3 Ï

}

x 5 7

( 3 Ï}

x )3 5 73

x 5 343

Check: 3 Î}

343 2 10 0 23

7 2 10 0 23

23 5 23 ✓

14. 3 Î}

x 2 16 5 2

( 3 Î}

x 2 16 )3 5 23

x 2 16 5 8

x 5 24

Check: 3 Î}

24 2 16 0 2

3 Î}

8 0 2

2 5 2 ✓

15. 3 Î}

12x 2 13 5 27

3 Î}

12x 5 6

( 3 Î}

12x )3 5 63

12x 5 216

x 5 18

Check: 3 Î} 12(18) 2 13 0 27

3 Î}

216 2 13 0 27

6 2 13 0 27

27 5 27 ✓

16. 3 3 Î}

16x 2 7 5 17

3 3 Î}

16x 5 24

3 Î}

16x 5 8

( 3 Î}

16x )3 5 83

16x 5 512

x 5 32

Check: 3 3 Î} 16(32) 2 7 0 17

3 3 Î}

512 2 7 0 17

24 2 7 0 17

17 5 17 ✓

17. 25 3 Î}

8x 1 12 5 28

25 3 Î}

8x 5 220

3 Î}

8x 5 4

( 3 Î}

8x )3 5 43

8x 5 64

x 5 8

Check: 25 3 Î}

8(8) 1 12 0 28

25 3 Î}

64 1 12 0 28

220 1 12 0 28

28 5 28 ✓

18. 3 Î}

4x 1 5 5 1 } 2

( 3 Î}

4x 1 5 )3 5 1 1 } 2 2 3

4x 1 5 5 1 } 8

4x 5 2 39

} 8

x 5 2 39

} 32

Check: 3 Î} 4 1 39 } 32 2 1 5 0 1 }

2

3

Î} 2 39

} 8 1 5 0 1 } 2

3

Î}

1 }

8 0 1 }

2

1 }

2 5

1 } 2 ✓

19. 3 Î}

x 2 3 1 2 5 4

3 Î}

x 2 3 5 2

( 3 Î}

x 2 3 )3 5 23

x 2 3 5 8

x 5 11

Check: 3 Î}

11 2 3 1 2 0 4

3 Î}

8 1 2 0 4

4 5 4 ✓

20. 3 Î}

4x 1 2 2 6 5 210

3 Î}

4x 1 2 5 24

( 3 Î}

4x 1 2 )3 5 (24)3

4x 1 2 5 264

4x 5 266

x 5 2 33

} 2

Check: 3

Î} 4 1 2 33

} 2 2 1 2 2 6 0 210

3 Î}

266 1 2 2 6 0 210

3 Î}

264 2 6 0 210

24 2 6 0 210

210 5 210 ✓

21. 24 3 Î}

x 1 10 1 3 5 15

24 3 Î}

x 1 10 5 12

3 Î}

x 1 10 5 23

( 3 Î}

x 1 10 )3 5 (23)3

x 1 10 5 227

x 5 237

Check: 24 3 Î}

237 1 10 1 3 0 15

24 3 Î}

227 1 3 0 15

12 1 3 0 15

15 5 15 ✓


n2ws-06b.indd 394 6/27/06 10:40:21 AM

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395Algebra 2


22. Sample answer: 3 Î}

23x 2 1 5 2

Substitute 23 for x in the equation 3 Î}

ax 1 b 5 c and fi nd values of a, b, and c that satisfy the equation. Let c 5 2.

3 Î} a(23) 1 b 5 2

3 Î}

8 5 2

23a 1 b 5 8

23(23) 2 1 5 8

a 5 23, b 5 21, c 5 2

23. 2x2/3 5 32

x2/3 5 16

(x2/3)3/2 5 163/2

x 5 6(161/2)3

x 5 6(4)3

x 5 664


2(64)2/3 0 32 2(264)2/3 0 32

2(16) 0 32 2(16) 0 32

32 5 32 ✓ 32 5 32 ✓

24. 1 }

2 x5/2 5 16

x5/2 5 32

(x5/2)2/5 5 322/5

x 5 4

Check: 1 }

2 (45/2) 0 16

1 }

2 (32) 0 16

16 5 16 ✓

25. 9x2/5 5 36

x2/5 5 4

(x2/5)5/2 5 45/2

x 5 6(41/2)5

x 5 6(2)5

x 5 632


9(32)2/5 0 36 9(232)2/5 0 36

9(4) 0 36 9(4) 0 36

36 5 36 ✓ 36 5 36 ✓

26. (8x)4/3 1 44 5 300

(8x)4/3 5 256

[(8x)4/3]3/4 5 2563/4

8x 5 6(2561/4)3

8x 5 6(4)3

8x 5 664

x 5 68


[8(8)]4/3 1 44 0 300 [8(28)]4/3 1 44 0 300

644/3 1 44 0 300 (264)4/3 1 44 0 300

256 1 44 0 300 256 1 44 0 300

300 5 300 ✓ 300 5 300 ✓

27. 1 } 7 (x 1 9)3/2 5 49

(x 1 9)3/2 5 343

[(x 1 9)3/2]2/3 5 3432/3

x 1 9 5 49

x 5 40

Check: (40 1 9)3/2 0 343

493/2 0 343

343 5 343 ✓

28. (x 2 5)5/3 2 73 5 170

(x 2 5)5/3 5 243

[(x 2 5)5/3]3/5 5 2433/5

x 2 5 5 27

x 5 32

Check: (32 2 5)5/3 2 73 0 170

275/3 2 73 0 170

243 2 73 0 170

170 5 170 ✓

29. 1 1 } 3 x 2 11 2 1/2

5 5

F 1 1 } 3 x 2 11 2 1/2

G 2 5 52

1 }

3 x 2 11 5 25

1 }

3 x 5 36

x 5 108

Check: F 1 } 3 (108) 2 11 G 1/2 0 5

(36 2 11)1/2 0 5

251/2 0 5

5 5 5 ✓

30. (5x 2 19)5/6 5 32

[(5x 2 19)5/6]6/5 5 326/5

5x 2 19 5 64

5x 5 83

x 5 83

} 5

Check: F 5 1 83 } 5 2 2 19 G 5/6

0 32

(83 2 19)5/6 0 32

645/6 0 32

32 5 32 ✓


n2ws-06b.indd 395 6/27/06 10:40:24 AM

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31. (3x 1 43)2/3 1 22 5 38

(3x 1 43)2/3 5 16

[(3x 1 43)2/3]3/2 5 163/2

3x 1 43 5 6(161/2)3

3x 1 43 5 6(4)3

3x 1 43 5 664

3x 5 243 6 64

x 5 7 or x 5 2 107

} 3

Check x 5 7: Check x 5 2 107

} 3 :

[3(7) 1 43]2/3 1 22 0 38 F 3 1 2 107

} 3 2 1 43 G 2/3 1 22 0 38

642/3 1 22 0 38 (264)2/3 1 22 0 38

16 1 22 0 38 16 1 22 0 38

38 5 38 ✓ ✓ 38 5 38

32. Before each side of the equation was cubed, the radical should have been isloated on the one side of the equation.

3 Ï

}

x 1 2 5 4

3 Ï

}

x 5 2

( 3 Ï}

x )3 5 23

x 5 8

33. Each side of the equation should have been squared, no just one side.

(x 1 7)1/2 5 5

[(x 1 7)1/2]2 5 52

x 1 7 5 25

x 5 18

34. x 2 6 5 Ï}

3x

(x 2 6)2 5 ( Ï}

3x )2

x2 2 12x 1 36 5 3x

x2 2 15x 1 36 5 0

(x 2 12)(x 2 3) 5 0

x 2 12 5 0 or x 2 3 5 0

x 5 12 or x 5 3


12 2 6 0 Ï}

3(12) 3 2 6 0 Ï}

3(3)

6 0 Ï}

36 3 2 6 0 Ï}

9

6 5 6 ✓ 23 Þ 3


35. x 2 10 5 Ï}

9x

(x 2 10)2 5 ( Ï}

9x )2

x2 2 20x 1 100 5 9x

x2 2 29x 1 100 5 0

(x 2 4)(x 2 25) 5 0

x 2 4 5 0 or x 2 25 5 0

x 5 4 or x 5 25


4 2 10 0 Ï}

9(4) 25 2 10 0 Ï}

9(25)

26 0 Ï}

36 15 0 Ï}

255

26 Þ 6 15 5 15


36. x 5 Ï}

16x 1 225

x2 5 ( Ï}

16x 1 225 )2

x2 5 16x 1 225

x2 2 16x 2 225 5 0

(x 2 25)(x 1 9) 5 0

x 2 25 5 0 or x 1 9 5 0

x 5 25 or x 5 29


25 0 Ï}}

16(25) 1 255 29 0 Ï}}

16(29) 1 225

25 0 Ï}

400 1 225 29 0 Ï}}

2144 1 225

25 0 Ï}

625 29 0 Ï}

81

25 5 25 ✓ 29 Þ 9


37. Ï}

21x 1 1 5 x 1 5

( Ï}

21x 1 1 )2 5 (x 1 5)2

21x 1 1 5 x2 1 10x 1 25

0 5 x2 2 11x 1 24

0 5 (x 2 8)(x 2 3)

x 2 8 5 0 or x 2 3 5 3

x 5 8 or x 5 3


Ï}

21(8) 1 1 0 8 1 5 Ï}

21(3) 1 1 0 3 1 5

Ï}

169 0 13 Ï}

64 0 8

13 5 13 ✓ 8 5 8 ✓



n2ws-06b.indd 396 6/27/06 10:40:28 AM

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397Algebra 2


38. Ï}

44 2 2x 5 x 2 10

( Ï}

44 2 2x )2 5 (x 2 10)2

44 2 2x 5 x2 2 20x 1 100

0 5 x2 2 18x 1 56

0 5 (x 2 4)(x 2 14)

x 2 4 5 0 or x 2 14 5 0

x 5 4 or x 5 14


Ï}

44 2 2(4) 0 4 2 10 Ï}

44 2 2(14) 0 14 2 10

Ï}

36 0 26 Ï}

16 0 4

6 Þ 26 4 5 4 ✓


39. Ï}

x2 1 4 5 x 1 5

( Ï}

x2 1 4 )2 5 (x 1 5)2

x2 1 4 5 x2 1 10x 1 25

221 5 10x

2 21

} 10 5 x

Check: Î} 1 2 21

} 10 2 2 1 4 0 2 21

} 10 1 5

Ï}

8.41 0 2.9

2.9 5 2.9 ✓

40. x 2 2 5 Ï}

3 }

2 x 2 2

(x 2 2)2 5 1 Ï}

3 }

2 x 2 2 2 2

x2 2 4x 1 4 5 3 } 2 x 2 2

x2 2 11

} 2 x 1 6 5 0

(x 2 4) 1 x 2 3 } 2 2 5 0

x 2 4 5 0 or x 2 3 } 2 5 0

x 5 4 or x 5 3 } 2


4 2 2 0 Î} 3 }

2 (4) 2 2

3 }

2 2 2 0 Î}

3 }

2 1 3 }

2 2 2 2

2 0 Ï}

4 2 1 } 2 0 Î}

9 }

4 2 2

2 5 2 ✓ 2 1 } 2 0 Î}

1 }

4

2 1 } 2 Þ

1 } 2


41. 4 Î}

3 2 8x2 5 2x

( 4 Ï}

3 – 8x2 )4 5 (2x)4

3 2 8x2 5 16x4

0 5 16x4 1 8x2 2 3

0 5 (4x2 1 3)(4x2 2 1) 0 5 (4x2 1 3)(2x 1 1)(2x 2 1)

4x2 1 3 5 0 or 2x 1 1 5 0 or 2x 2 1 5 0

x2 5 2 3 } 4 2x 5 21 2x 5 1

x 5 6 Ï}

2 3 } 4 x 5 2

1 } 2 x 5

1 } 2

x 5 6 i Ï

}

3 } 2

Check x 5 i Ï

}

3 } 2 :

4

Î}}

3 2 8 1 i Ï}

3 } 2 2

2

0 2 1 i Ï}

3 } 2 2

4

Î} 3 2 8 1 i2 Ï}

9 }

4 2 0 i Ï

}

3

4 Î} 3 1 2(3) 0 i Ï

}

3

4 Î}

9 Þ i Ï}

3

Check x 5 2 i Ï

}

3 } 2 :

4

Î}}

3 2 8 1 2 i Ï

}

3 } 2 2

2

0 2 1 2 i Ï

}

3 } 2 2

4 Î} 3 2 8 1 i2 Ï}

9 }

4 2 0 2i Ï

}

3

4 Î} 3 1 2(3) 0 2i Ï

}

3

4 Ï}

9 Þ 2i Ï}

3

Check x 5 2 1 } 2 :

4

Î} 3 2 8 1 2 1 } 2 2 2 0 2 1 2

1 } 2 2

4 Î} 3 2 8 1 1 }

4 2 0 21

4 Î}

3 2 2 0 21

1 Þ 21

Check x 5 1 } 2 :

4 Î} 3 2 8 1 1 }

2 2 2 0 2 1 1 }

2 2

4 Î} 3 2 8 1 1 }

4 2 0 1

4 Î}

3 2 2 0 1

1 5 1 ✓

The only solution is 1 }

2 .


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42. 3 Î}

8x3 2 1 5 2x 2 1

( 3

Î}

8x3 2 1 )3 5 (2x 2 1)3

8x3 2 1 5 (2x 2 1)(2x 2 1)(2x 2 1)

8x3 2 1 5 (4x2 2 4x 1 1)(2x 2 1)

8x3 2 1 5 8x3 2 8x2 1 2x 2 4x2 1 4x 2 1

12x2 2 6x 5 0

6x(2x 2 1) 5 0

6x 5 0 or 2x 2 1 5 0

x 5 0 or x 5 1 } 2


3 Î} 8(0)3 2 1 0 2(0) 2 1

3

Î} 8 1 1 } 2 2 3 2 1 0 2 1 1 }

2 2 2 1

3 Î}

21 0 21 3

Î} 8 1 1 } 8 2 2 1 0 0

21 5 21 ✓ 3 Î}

0 0 0

0 5 0 ✓

The solutions are 0 and 1 }

2 .

43. A;

Ï}

32x 2 64 5 2x

( Ï}

32x 2 64 )2 5 (2x)2

32x 2 64 5 4x2

0 5 4x2 2 32x 1 64

0 5 4(x2 2 8x 1 16) 0 5 4(x 2 4)(x 2 4)

x 2 4 5 0

x 5 4

44. You can tell that Ï}

x 1 4 5 25 has no solution because the positive square root of a number is never negative.

45. Ï}

4x 1 1 5 Ï}

x 1 10

( Ï}

4x 1 1 )2 5 ( Ï

}

x 1 10 )2

4x 1 1 5 x 1 10

3x 5 9

x 5 3

Check: Ï}

4(3) 1 1 0 Ï}

3 1 10

Ï}

13 5 Ï}

13 ✓

46. 3 Î}

12x 2 5 2 3 Î}

8x 1 15 5 0

3 Î}

12x 2 5 5 3 Î}

8x 1 15

( 3 Î}

12x 2 5 )3 5 ( 3

Î}

8x 1 15 )3

12x 2 5 5 8x 1 15

4x 5 20

x 5 5

Check: 3 Î} 12(5) 2 5 2 3

Î} 8(5) 1 15 0 0

3 Î}

55 2 3 Î}

55 0 0

0 5 0 ✓

47. Ï}

3x 2 8 1 1 5 Ï}

x 1 5

( Ï}

3x 2 8 1 1)2 5 ( Ï

} x 1 5 )2

3x 2 8 1 2 Ï}

3x 2 8 1 1 5 x 1 5

2 Ï}

3x 2 8 5 22x 1 12

Ï}

3x 2 8 5 2x 1 6

( Ï}

3x 2 8 )2 5 (2x 1 6)2

3x 2 8 5 x2 2 12x 1 36

0 5 x2 2 15x 1 44

0 5 (x 2 4)(x 2 11)

x 2 4 5 0 or x 2 11 5 0

x 5 4 or x 5 11

Check: x 5 4: Check x 5 11:

Ï}

3(4) 2 8 1 1 0 Ï}

4 1 5 Ï}

3(11) 2 8 1 1 0 Ï}

11 1 5

Ï}

4 1 1 0 Î}

9 Ï}

3(11) 2 8 1 1 0 Ï}

16

3 5 3 ✓ 6 Þ 4


48. Ï}

2 }

3 x 2 4 5 Ï

}

2 } 5 x 2 7

1 Ï}

2 }

3 x 2 4 2 2 5 1 Ï

}

2 } 5 x 2 7 2 2

2 }

3 x 2 4 5

2 } 5 x 2 7

4 }

15 x 5 23

x 5 2 45

} 4

Check: Î} 2 }

3 1 2

45 } 4 2 2 4 0 Ï

}

2 } 5 1 2

45 } 4 2 2 7

Ï}

2 15

} 2 2 4 0 Ï}

2 9 } 2 2 7

Ï}

2 23

} 2 5 Ï}

2 23

} 2 ✓

49. Ï}

x 1 2 5 2 2 Ï} x

( Ï}

x 1 2 )2 5 (2 2 Ï

}

x )2

x 1 2 5 4 2 4 Ï}

x 1 x

4 Ï}

x 5 2

Ï}

x 5 1 } 2

( Ï}

x )2 5 1 1 } 2 2 2

x 5 1 } 4

Check: Ï}

1 }

4 1 2 0 2 2 Ï

}

1 }

4

Ï}

9 }

4 0 2 2

1 } 2

3 }

2 5

3 } 2 ✓


n2ws-06b.indd 398 6/27/06 10:40:38 AM

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399Algebra 2


50. Ï}

2x 1 3 1 2 5 Ï}

6x 1 7

( Ï}

2x 1 3 1 2)2 5 ( Ï

}

6x 1 7 )2

2x 1 3 1 4 Ï}

2x 1 3 1 4 5 6x 1 7

4 Î}

2x 1 3 5 4x

Ï}

2x 1 3 5 x

( Ï}

2x 1 3 )2 5 x2

2x 1 3 5 x2

0 5 x2 2 2x 2 3

0 5 (x 2 3)(x 1 1)

x 2 3 5 0 or x 1 1 5 0

x 5 3 or x 5 21

Check x 5 3:

Ï}

2(3) 1 3 1 2 0 Ï}

6(3) 1 7

Ï}

9 1 2 0 Ï}

25

5 5 5 ✓

Check x 5 21:

Ï}

2(21) 1 3 1 2 0 Ï}

6(21) 1 7

Ï}

1 1 2 0 Ï}

1

3 Þ 1


51. Ï}

2x 1 5 5 Ï}

x 1 2 1 1

( Ï}

2x 1 5 )2 5 ( Ï

}

x 1 2 1 1)2

2x 1 5 5 x 1 2 1 2 Ï}

x 1 2 1 1

x 1 2 5 2 Ï}

x 1 2

(x 1 2)2 5 (2 Ï}

x 1 2 )2

x 2 1 4x 1 4 5 4(x 1 2)

x2 1 4x 1 4 5 4x 1 8

x2 5 4

x 5 6 Ï}

4 5 62

Check x 5 2:

Ï}

2(2) 1 5 0 Ï}

2 1 2 1 1

Ï}

9 0 Ï}

4 1 1

3 5 3 ✓

Check x 5 22:

Ï}

2(22) 1 5 0 Ï}

22 1 2 1 1

Ï}

1 0 Ï}

0 1 1

1 5 1 ✓


52. Ï}

5x 1 6 1 3 5 Ï}

3x 1 3 1 4

Ï}

5x 1 6 5 Ï}

3x 1 3 1 1

( Ï}

5x 1 6 )2 5 ( Ï

}

3x 1 3 1 1)2

5x 1 6 5 3x 1 3 1 2 Ï}

3x 1 3 1 1

2x 1 2 5 2 Ï}

3x 1 3

x 1 1 5 Ï}

3x 1 3

(x 1 1)2 5 ( Ï}

3x 1 3 )2

x2 1 2x 1 1 5 3x 1 3

x2 2 x 2 2 5 0

(x 2 2)(x 1 1) 5 0

x 2 2 5 0 or x 1 1 5 0

x 5 2 or x 5 21

Check x 5 2:

Ï}

5(2) 1 6 1 3 0 Ï}

3(2) 1 3 1 4

Ï}

16 1 3 0 Ï}

9 1 4

7 5 7 ✓

Check x 5 21:

Ï}

5(21) 1 6 1 3 0 Ï}

3(21) 1 3 1 4

Ï}

1 1 3 0 Ï}

0 1 4

4 5 4 ✓


53. 3 Ï}

x 1 5 Ï}

y 5 31

5 Ï}

x 2 5 Ï}

y 5 215

8 Ï}

x 5 16

Ï}

x 5 2

( Ï}

x )2 5 22

x 5 4

3 Ï}

4 1 5 Ï}

y 5 31

6 1 5 Ï}

y 5 31

5 Ï}

y 5 25

Ï}

y 5 5

( Ï}

y )2 5 52

y 5 25

The solution is (4, 25).

54. 5 Ï}

x 2 2 Ï}

y 5 4 Ï}

2 3 3 15 Ï}

x 2 6 Ï}

y 5 12 Ï}

2

2 Ï}

x 1 3 Ï}

y 5 13 Ï}

2 3 2 4 Ï}

x 1 6 Ï}

y 5 26 Ï}

2

19 Ï}

x 5 38 Ï}

2

Ï}

x 5 2 Ï}

2

( Ï}

x )2 5 (2 Ï}

2 )2

x 5 8

5 Ï}

8 2 2 Ï}

y 5 4 Ï}

2

10 Ï}

2 2 2 Ï}

y 5 4 Ï}

2

22 Ï}

y 5 26 Ï}

2

Ï}

y 5 3 Ï}

2

( Ï}

y )2 5 (3 Ï

}

2 )2

y 5 18



n2ws-06b.indd 399 6/27/06 10:40:43 AM

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55. Sample answer: Ï}

4x 2 3 5 2x

( Ï}

4x 2 3 )2 5 (2x)2

4x 2 3 5 x2

0 5 x2 2 4x 1 3

0 5 (x 2 1)(x 2 3)

x 5 1 or x 5 3


Ï}

4(1) 2 3 0 21 Ï}

4(3) 2 3 0 23

1 Þ 21 3 Þ 23

There is no solution.

Problem Solving

56. v 5 Ï}

2gh

15 5 Ï}

2(9.8)h

15 5 Ï}

19.6h

152 5 ( Ï}

19.6h )2

225 5 19.6h

11.5 ø h The height at the top of the swing was about 11.5 meters.

57. r 5 Ï}

kt } π(h0 2 h)

0.875 5 Ï}

0.04t } π(6.5 2 0)

0.875 5 Ï}

0.04t }

Ï}

6.5π�

0.875 ø Ï}

0.04t }

4.5

4 ø Ï}

0.04t

42 ø ( Ï}

0.04t )2

16 ø 0.04t

400 ø t

It will take about 400 minutes for the entire candle to burn.

58. l 5 54d3/2

3 5 54d3/2

1 }

18 5 d3/2

1 1 } 18

2 2/3 5 (d 3/2)2/3

0.15 ø d The diameter of the nail is about 0.15 inch.

59. h 5 62.5 3 Î}

t 1 75.8

Elephant with h 5 150: Elephant with h 5 250:

150 5 62.5 3 Î}

t 1 75.8 250 5 62.5 3 Î}

t 1 75.8

74.2 5 62.5 3 Î}

t 174.2 5 62.5 3 Î}

t

1.2 ø 3 Î}

t 2.8 ø 3 Î}

t

1.23 ø ( 3 Î}

t )3 2.83 ø ( 3 Î}

t )3

2 ø t 22 ø t The elephant with a shoulder height of 250 centimeter

is about 20 years older than the elephant with a shoulder height of 150 centimeters.

60. a. t 5 0.5 Ï}

h

Basketball player: Kangaroo:

0.81 5 0.5 Ï}

h 1.12 5 0.5 Ï}

h

1.62 5 Ï}

h 2.24 5 Ï}

h

1.622 5 ( Ï}

h )2 2.242 5 ( Ï

}

h )2

2.6 ø h 5 ø h The basketball player The kangaroo

jumped a height of jumped a heightabout 2.6 feet. of about 5 feet.

b. Basketball player: Kangaroo:

2(0.81) 5 0.5 Ï}

h 2(1.12) 5 0.5 Ï}

h

3.24 5 Ï}

h 4.48 5 Ï}

h

10.5 ø h 4.482 5( Ï}

h )2

20 ø h c. No; if the hang time doubles, the height jumped does

not double, it quadruples. To find the new height jumped, you should double the hang time and square the results, which means that the new height will be 22 5 4 times greater than the old height.

61. a. B 5 1.69 Ï}

s 1 4.25 2 3.55

0 5 1.69 Ï}

s 1 4.25 2 3.55

3.55 5 1.69 Ï}

s 1 4.25

2.12 ø ( Ï}

s 1 4.25 )2

4.41 ø s 1 4.25

0.16 ø s The windspeed is about 0.16 mile per hour.

b. B 5 1.69 Ï}

s 1 4.25 2 3.55

12 5 1.69 Ï}

s 1 4.25 2 3.55

15.55 5 1.69 Ï}

s 1 4.25

9.2 ø Ï}

s 1 4.25

9.22 ø ( Ï}

s 1 4.25 )2

84.64 ø s 1 4.25

80.39 ø s The windspeed is about 80.39 miles per hour.

c. 0.16 ≤ s ≤ 80.39


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401Algebra 2


62. l 5 Ï}}

h2 1 1 } 4 (b2 2 b1)2

5 5 Ï}}

h2 1 1 } 4 (4 2 2)2

5 5 Ï}

h2 1 1

52 5 1 Ï}

h2 1 1 2 2

25 5 h2 1 1

24 5 h2

Ï}

24 5 h

2 Ï}

6 5 h

4.9 ø h

The height of the pyramid is 4.9 units.

Mixed Review

63. 43 p 42 5 43 1 2 Product of powers property

5 45

5 1024

64. (322)3 5 322 p 3 Power of a power property

5 3 2 6

5 1 }

36 Negative exponent property

5 1 } 729

65. (25)(25)24 5 (25)1 1 (24) Product of powers property

5 (25)23

5 1 }

(25)3 Negative exponent property

5 2 1 } 125

66. (1023)21 5 1023 p (21) Power of a power property

5 103

5 1000

67. 824 p 83 5 824 1 3 Product of powers property

5 821

5 1 } 8 Negative exponent property

68. 60 p 64 p 624 5 60 1 4 1 (24) Product of powers property

5 60

5 1 Zero exponent property

69. f (x) 5 2x3

x 23 22 21 0 1 2 3

f (x) 27 8 1 0 21 28 227

1

x

y

21

70. f (x) 5 x4 2 9

x 23 22 21 0 1 2 3

f(x ) 72 7 28 29 28 7 72

2

x

y

1

71. f (x) 5 x3 1 2

x 23 22 21 0 1 2 3

f(x) 225 26 1 2 3 10 29

4

x

y

1

72. f (x) 5 x4 2 8x2 2 48

x 23 22 21 0 1 2 3

f (x) 239 264 255 248 255 264 239

10

x

y

21

73. f (x) 5 2 1 } 3 x3 1 x

x 26 23 0 3 6

f (x) 66 6 0 26 266

2

x

y

1


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74. f (x) 5 x5 2 2x 2 4

x 22 21 0 1 2

f (x) 232 23 24 25 24

1

x

y

21

75. 163/2 5 ( Ï}

16 )3 5 43 5 64

76. 1 }

825/3 5 85/3 5 ( 3 Î}

8 )5 5 25 5 32

77. 22561/4 5 2 4 Î}

256 5 24

78. 425/2 5 1 }

45/2 5 1 }

1 Ï}

4 2 5 5

1 }

25 5 1 } 32

79. 312523/5 5 1 }

31253/5 5 1 }

( 5 Î}

3125 )3 5

1 }

53 5 1 } 125

80. 1 }

274/3 5 1 }

( 3 Î}

27 )4 5 1 }

34 5 1 } 81

Quiz 6.5–6.6 (p. 459)

1. y 5 4 Ï}

x

x 0 1 2 3 4

y 0 4 5.66 6.93 8

1

x

y

21


2. y 5 Ï}

x 1 3

1

x

y

21


x up 3 units.

The domain is x ≥ 0 and range is y ≥ 3.

3. g(x) 5 Ï}

x 1 2 2 5

1

x

y

21

Because h 5 22 and k 5 25, shift that graph of g(x) 5 Ï

}


The domain is x ≥ 22 and y ≥ 25.

4. y 5 2 1 } 2

3 Î}

x

x 22 21 0 1 2

y 0.63 0.5 0 20.5 20.63

1

x

y

22


5. f (x) 5 3 Ï}

x 2 4

1

x

y

21


6. y 5 3 Î}

x 2 3 1 2

1

x

y

21


x to the right 3 units and up 2 units. The domain and range are both all real numbers.

7. Ï}

6x 1 15 5 9

( Ï}

6x 1 15 )2 5 92

6x 1 15 5 81

6x 5 66

x 5 11

Check: Ï}

6(11) 1 15 0 9

Ï}

81 0 9

9 5 9 ✓


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403Algebra 2


8. 1 }

4 (7x 1 8)3/2 5 54

(7x 1 8)3/2 5 216

[(7x 1 8)3/2]2/3 5 2162/3

7x 1 8 5 36

7x 5 28

x 5 4

Check: 1 }

4 [7(4) 1 8]3/2 0 54

1 }

4 (36)3/2 0 54

1 }

4 (216) 0 54

54 5 54 ✓

9. 3 Î}

3x 1 5 1 2 5 5

3 Î}

3x 1 5 5 3

( 3 Î}

3x 1 5 )3 5 33

3x 1 5 5 27

3x 5 22

x 5 22

} 3

Check: 3 Î} 3 1 22 }

3 2 1 5 1 2 0 5

3 Î}

22 1 5 1 2 0 5

3 Î}

27 1 2 0 5

5 5 5 ✓

10. x 2 3 5 Ï}

10x 2 54

(x 2 3)2 5 ( Ï}

10x 2 54 )2

x2 2 6x 1 9 5 10x 2 54

x2 2 16x 1 63 5 0

(x 2 9)(x 2 7) 5 0

x 2 9 5 0 or x 2 7 5 0

x 5 9 or x 5 7


9 2 3 0 Ï}

10(9) 2 54 7 2 3 0 Ï}

10(7) 2 54

6 0 Ï}

36 4 0 Ï}

16

6 5 6 ✓ 4 5 4 ✓


11. Ï}

4x 2 4 5 Ï}

5x 2 1 2 1

( Ï}

4x 2 4 )2 5 ( Ï}

5x 2 1 2 1)2

4x 2 4 5 5x 2 1 2 2 Ï}

5x 2 1 1 1

2 Ï}

5x 2 1 5 x 1 4

(2 Ï}

5x 2 1 )2 5 (x 1 4)2

4(5x 2 1) 5 x2 1 8x 1 16

20x 2 4 5 x2 1 8x 1 16

0 5 x2 2 12x 1 20

0 5 (x 2 10)(x 2 2)

x 2 10 5 0 or x 2 2 5 0

x 5 10 or x 5 2

Check x 5 10:

Ï}

4(10) 2 4 0 Ï}

5(10) 2 1 2 1

Ï}

36 0 Ï}

49 2 1

6 5 6 ✓

Check x 5 2:

Ï}

4(2) 2 4 0 Ï}

5(2) 2 1 2 1

Ï}

4 0 Ï}

9 2 1

2 5 2 ✓


12. 3 Î} 4 } 5 x 2 9 5 3

Î}

x 2 6

1 3 Î} 4 } 5 x 2 9 2 3 5 ( 3 Ï

}

x 2 6 )3

4 } 5 x 2 9 5 x 2 6

23 5 1 } 5 x

215 5 x

Check: 3 Î}}

4 } 5 (215) 2 9 0

3 Î}

215 2 6

3 Î}

212 2 9 0 3 Î}

221

3 Î}

221 5 3 Î}

221 ✓

The solution is 215.

13. P 5 0.199a3/2

(1.88 years) 1 365 days }

1year 2 5 686.2 days

686.2 5 0.199a3/2

3448 ø a3/2

34482/3 ø (a3/2)2/3

228 ø a The length of the orbit’s minor axis is about

228 million kilometers.

Problem Solving Workshop 6.6 (p. 461)

1. Ï}

25 2 x 5 8

X Y1 -40 8.0623 -39 8 -38 7.9373 -37 7.874 -36 7.8102 -35 7.746-34 7.6811X=-39

IntersectionX=-39 Y=8

The solution is 239.


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2. 2.3 Ï}

x 2 1 5 11.5

X Y1 25 11.268 26 11.5 27 11.728 28 11.951 29 12.17 30 12.38631 12.598X=26

IntersectionX=26 Y=11.5

The solution is 26.

3. 4.3 Ï}

x 2 7 5 30

X Y1 55.5 29.946 55.6 29.977 55.7 30.008 55.8 30.039 55.9 30.06956.0 30.156.1 30.131X=55.7

IntersectionX=55.7 Y=30

The solution is about 55.7.

4. 6 Ï}

2 2 7x 2 1.2 5 22.8

X Y1 -6 38.599 -5 35.297 -4 31.663 -3 27.575-2 22.8-1 16.80 7.2853X=-2

IntersectionX=-2 Y=22.8

The solution is 22.

5. d 5 Ï}

625 1 h2

100 5 Ï}

625 1 h2

X Y1 95 98.234 96 99.202 97 100.17 98 101.1499 102.11100 103.08101 104.05X=97

IntersectionX=97 Y=100

The rocket’s height is about 97 feet.

6. v( p) 5 6.3 Ï}

1013 2 p

25 5 6.3 Ï}

1013 2 p

X Y1 995 26.729 996 25.976 997 25.2 998 24.4999 23.5721000 22.7151001 21.824X=997


The mean sustained wind velocity is 25 meters per second when the air pressure is about 997 millibars.

7. L 5 πr Ï}

r2 1 h2

900 5 π(7.5) Ï}

(7.5)2 1 h2

900 5 7.5π Ï}

56.25 1 h2

X Y1 35 843.39 36 866.44 37 889.52 38 912.6339 935.7540 958.941 952.07X=37


The height of the cone is about 37 centimeters.

6.6 Extension (p.463)

1. 2 Ï}

x 2 5 ≥ 3

X Y1 15 2.746 16 3 17 3.246218 3.485319 3.717820 3.944321 4.1652X=16

X Y1 -3 ERROR -2 ERROR -1 ERROR0 -51 -32 -2.1723 -1.536X=0

The domain is x ≥ 0.

The solution of the inequality is x ≥ 16.

2. Ï}

x 2 4 ≤ 5

X Y1 25 4.5826 26 4.6904 27 4.795828 4.89929 530 5.09931 5.1962X=29

X Y1 0 ERROR 1 ERROR 2 ERROR3 ERROR4 05 16 1.4142X=4


The solution of the inequality is 4 ≤ x ≤ 29.

3. 4 Ï}

x 1 1 ≤ 9

X Y1 -2 ERROR -1 ERROR 0 11 52 6.65693 7.92824 9X=4


The solution is the inequality 0 ≤ x ≤ 4.

4. Ï}

x 1 7 ≥ 3

X Y1 0 2.6458 1 2.8284 2 33 3.16234 3.31665 3.46416 3.6056X=2

X Y1 -10 ERROR -9 ERROR -8 ERROR-7 0-6 1-5 2-4 3X=-7




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405Algebra 2


5. Ï}

x 1 Ï}

x 1 3 ≥ 3

X Y1 -3 ERROR -2 ERROR -1 ERROR0 1.73211 3 2 3.65033 4.1815X=1



6. Ï}

x 1 Ï}

x 2 5 ≤ 5

X Y1 3 ERROR 4 ERROR 5 2.23616 3.44957 4.06 8 4.56059 5X=9


The solution of the inequality is 5 ≤ x ≤ 9.

7. 2 Ï}

x 1 3 ≤ 8


The domain of y 5 2 Ï}

x 1 3 is x ≥ 0.

The solution of the inequality is 0 ≤ x ≤ 6.25. 8. Ï

}

x 1 3 ≥ 2.6

IntersectionX=3.76 Y=2.6


The solution of the inequality is x ≥ 3.76. 9. 7 Ï

}

x 1 1 < 9



x 1 1 is x ≥ 0.

The solution of the inequality is approximately 0 ≤ x < 1.3.

10. 4 Ï}

3x 2 7 > 7.8

IntersectionX=3.6 Y=7.8


3x 2 7 is x ≥ 7 } 3 .

The solution of the inequality is approximately x > 3.6.

11. Ï}

x 2 Ï}

x 1 5 < 21

IntersectionX=4 Y=-1

The domain of y 5 Ï}

x 2 Ï}

x 1 5 is x ≥ 0.

The solution of the inequality is 0 ≤ x < 4.

12. Ï}

x 1 2 1 Ï}

x 2 1 ≤ 9


The domain of y 5 Ï}

x 1 2 1 Ï}

x 2 1 is x ≥ 1. The solution of the inequality is approximately is

1 ≤ x ≤ 19.8.

13. l 1 1.25 Ï}

s 2 9.8 3 Î}

d ≤ 16

20 1 1.25 Ï}

s 2 9.8 3 Î}

27 ≤ 16

20 1 1.25 Ï}

s 2 9.8(3) ≤ 16

20 1 1.25 Ï}

s 2 29.4 ≤ 16

1.25 Ï}

s 2 9.4 ≤ 16

X Y1 410 15.911 411 15.941 412 15.972413 16.003414 16.034415 16.064416 16.095X=413

X Y1 -2 ERROR -1 ERROR 0 -9.41 -8.152 -7.6323 -7.2354 -6.9X=0

The domain is s ≥ 0.

The solution of the inequality is approximately 0 ≤ s ≤ 413. The maximum allowable value for s is about 413 square meters.


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Mixed Review of Problem Solving (p. 464)

1. a. R 5 C 1 MC

R 5 C 1 0.4C

R 5 1.4C

b. R 5 1.4C

R }

1.4 5 C

c. C 5 R

} 1.4 5 60

} 1.4 ø 42.86

The cost of a pair of the women’s athletic shoes is $42.86.

2. Ï}

3x 2 5 5 4

( Ï}

3x 2 5 )2 5 42

3x 2 5 5 16

3x 5 21

x 5 7

You can use the graph of y 5 Ï}

3x 2 5 and the graph of y 5 4 to verify that x 5 7 is correct by fi nding thex-coordinate value of their intersection point.

3. Sample answer: Ï}

3x 1 19 5 2

Working Backwards:

x 5 25

3x 5 215

3x 1 19 5 4

Ï}

3x 1 19 5 2

4. a. k 5 0.134d

k }

0.134 5 d

The inverse function is d 5 k } 0.134 .

b. d 5 25 } 0.134 ø 186.57

You receive $186.57 for 25 kronor.

c. The function d 5 k } 0.134 gives U.S. dollar in terms of

Swedish kronor.

5. y 5 Ï}

x 1 14

(214, 0): 0 5 Ï}

214 1 14

(213, 1): 1 5 Ï}

213 1 14

Yes, there are other square root functions that pass through these points; two points do not detemine a unique square root function.

6. Sample answer: y 5 3 Ï}

x 2 4 , y 5 2 Ï} x 2 2

7. a. v 5 8 Ï}

h

v }

8 5 Ï

}

h

1 v } 8 2 2 5 ( Ï

}

h )2

v2

} 64

5 h

b. Sample answer: What is the maximum height of a toy rocket launched upward from ground level that has an initial velocity of 45 feet per second?

h 5 v2

} 64 5 452

} 64 5 2025

} 64 ø 31.6

The maximum height of the toy rocket is 31.6 feet.

8. d 5 Ï}

2500 1 h2

100 5 Ï}

2500 1 h2

1002 5 ( Ï}

2500 1 h2 )2

10,000 5 2500 1 h2

7500 5 h2

6 Ï}

7500 5 h

686.6 ø h Reject the negative value. The height of the balloon is

about 86.6 feet.

9. r(t) 5 6t, A(r) 5 πr2

A(r(t)) 5 π(6t)2 5 36πt2

A(r(2)) 5 36π(22) ø 452

The composition A(r(2)), or 452 square feet, represents the area of the outer ripple formed 2 seconds after the pebble hits the water.

Chapter 6 Review (pp. 466–468)

1. The index of the radical 4 Î}

7 is 4.

2. Sample answer: 3 Î}

6 , 8 3 Î}

6 ; 1 }

2 Ï

}

x , 2 3 } 5 Ï

}

x

3. Sample answer: A power function has the form y 5 axb where a is a real number and b is a rational number.

4. The graph of a function’s inverse is a refl ection of the graph of the original function in the line y 5 x.

5. The inverse of a function f is also a function if and only if no horizontal line intersects the graph of f more than once.

6. The graph of y 5 3 Î}

x 2 4 1 5 is a shift of the graph of

y 5 3 Î}

x 4 units right and 5 units up.

7. x2/3 5 5

(x2/3)3 5 53

x2 5 125

The student will have to take the square root of each side. To solve the equation in just one step, the student could

have raised each side of the equation to the power 3 }

2 .

8. 811/4 5 3 9. 01/3 5 0

10. 3 Î}

264 5 24 11. 3 Î}

125 5 5

12. 2563/4 5 (2561/4)3 5 43 5 64

13. 2722/3 5 1 }

272/3 5 1 }

(271/3)2 5

1 }

32 5 1 } 9

14. ( 3 Î}

8 )7 5 27 5 128

15. 1 }

( 5 Î}

232 )23 5 ( 5

Î}

232 )3 5 (22)3 5 28

16. 3 Î}

80 5 3 Î}

8 p 10 5 3 Î}

8 p 3 Î}

10 5 2 3 Î}

10


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407Algebra 2


17. (34 p 54)21/4 5 [(3 p 5)4]21/4 5 (154)21/4 5 1521 5 1 } 15

18. (25a10b16)1/2 5 251/2(a10)1/2(b16)1/2 5 5a5b8

19. Ï}

18x5y4

} 49xz3 5 Ï

}

18x4y4

} 49z3 5

Ï}

9 p 2x4y4 }

Ï}

49z3 5

3x2y2 Ï}

2 }

7z Ï} z

5 3x2y2 Ï

}

2 p Ï} z }}

7z Ï} z p Ï}

z 5

3x2y2 Ï}

2z }

7z2

20. f (x) 1 g(x) 5 (4x 2 6) 1 (x 1 8) 5 5x 1 2

21. f (x) 2 g(x) 5 (4x 2 6) 2 (x 1 8) 5 4x 2 6 2 x 2 8

5 3x 2 14

22. f (x) p g(x) 5 (4x 2 6)(x 1 8) 5 4x2 1 26x 2 48

23. f (g(x)) 5 f(x 1 8) 5 4(x 1 8) 2 6 5 4x 1 32 2 6

5 4x 1 26

24. y 5 1 } 3 x 1 4

x 5 1 } 3 y 1 4

x 2 4 5 1 } 3 y

3x 2 12 5 y

25. y 5 4x2 1 9, x ≥ 0 x 5 4y2 1 9

x 2 9 5 4y2

1 }

4 x 2

9 } 4 5 y2

Ï}

1 }

4 x 2

9 } 4 5 y

Ï}

x 2 9

} 4 5 y

Ï}

x 2 9 }

2 5 y

26. f (x) 5 x3 2 4

y 5 x3 2 4

x 5 y3 2 4

x 1 4 5 y3

3 Î}

x 1 4 5 y

f 21(x) 5 3 Î}

x 1 4

27. y 5 Ï}

x 1 3 1 5


x to the left 3 units and up 5 units.

1

x

y

21


28. y 5 3 Ï}

x 1 1 2 4


x to the left 1 unit and down 4 units.

1

x

y

21

The domain x ≥ 21 and the range is y ≥ 24.

29. y 5 3 Î}

x 2 4 2 5

Because h 5 4 and k 5 25 shift the graph of y 5 3 Ï}

x to the right 4 units and down 5 units.

1

x

y

22


30. 3 Î}

5x 2 4 5 2

( 3 Î}

5x 2 4 )3 5 23

5x 2 4 5 8

5x 5 12

x 5 12

} 5

Check: 3 Î} 5 1 12

} 5 2 24 0 2

3 Î}

12 2 4 0 2

3 Î}

8 0 2

2 5 2 ✓

The solution is 12

} 5 .

31. 3x3/4 5 24

x3/4 5 8

(x3/4)4/3 5 84/3

x 5 16

Check: 3(163/4) 0 24

3(8) 0 24

24 5 24 ✓

The solution is 16.


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32. Ï}

x2 2 10 5 Ï}

3x

( Ï}

x2 2 10 )2 5 1 Ï

}

3x 2 2

x2 2 10 5 3x

x2 2 3x 2 10 5 0

(x 2 5)(x 1 2) 5 0

x 2 5 5 0 or x 1 2 5 0

x 5 5 or x 5 22


Ï}

52 2 10 0 Ï}

3(5) The domain of

Ï}

25 2 10 0 Ï}

15 f (x) 5 Ï}

3x is x ≥ 0 and x 5 22 < 0, so x 5 22 is not a solution.

Ï}

15 5 Ï}

15 ✓

The solution is 5.

Chapter 6 Test (p. 469)

1. 21251/3 5 25 2. 321/5 5 2

3. 4 Î}

81 5 3 4. 3 Î}

27 5 3

5. 85/3 5 (81/3)5 5 25 5 32

6. 1623/2 5 1 }

163/2 5 1 }

(161/2)3 5 1 }

43 5 1 } 64

7. ( 3 Î}

227 )2 5 (23)2 5 9

8. ( 3 Î}

64 )24 5 1 }

( 3 Î}

64 )4 5 1 }

44 5 1 } 256

9. 3 Î}

88 5 3 Î}

8 p 11 5 3 Î}

8 p 3 Î}

11 5 2 3 Î}

11

10. 5 Î}

16 p 5 Î}

8 5 5 Î}

16 p 8 5 5 Î}

128 5 5 Î}

32 p 4 5 5 Î}

32 p 5 Î}

4

5 2 5 Î}

4

11. Ï}

12

} 49

5 Ï

}

12 }

Ï}

49 5

2 Ï}

3 } 7

12. 3 Î}

24 }

3 Î}

9 5

3 Î}

24 }

3 Î}

9 p

3 Î}

3 }

3 Î}

3 5

3 Î}

72 }

3 Î}

27 5

3 Î}

8 p 3 Î}

9 } 3 5

2 3 Î}

9 } 3

13. 3 Î} 64x4y2 5

3 Î} 64x3xy2 5 4x

3 Î}

xy2

14. 4 Î} 2x6y 8z 5

4 Î} 2x4x2y8z 5 xy2

4 Î}

2x 2z

15. 5 Î}

x6

} y4 5

5 Î}

x6 p y

} y4 p y

5 5 Î}

x6y

} y5 5

5 Î}

x5xy }

5 Î}

y5 5

x 5 Î}

xy } y

16. Ï}

75x5y6

} 36xz5 5 Î}

75x4y6 p z }

36z5 p z 5 Î}

75x4y6z }

36z6 5 Ï}

25 p 3x4y6z }

Ï}

36z6

5 5x2y3 Ï

}

3z }

6z3

17. f (x) 1 g(x) 5 (2x 1 9) 1 (3x 2 1) 5 5x 1 8

The functions f and g each have the same domain: all real numbers. So, the domain of f 1 g also consists of all real numbers.

18. f (x) 2 g(x) 5 (2x 1 9) 2 (3x 2 1) 5 2x 1 9 2 3x 1 1

5 2x 1 10


19. f (x) p g(x) 5 (2x 1 9)(3x 2 1) 5 6x2 1 25x 2 9

The functions f and g each have the same domain: all real numbers. So, the domain of f p g also consists of all real numbers.

20. f (x)

} g(x)

5 2x 1 9

} 3x 2 1

The functions f and g each have the same domain:

all real numbers. Because g 1 1 } 3 2 5 0, the domain of

f } g is

all real numbers except x 5 1 } 3 .

21. f (g(x)) 5 f (3x 2 1) 5 2(3x 2 1) 1 9

5 6x 2 2 1 9 5 6x 1 7

The functions f and g each have the same domain: all real numbers. So, the domain of f ( g (x))also consists of all real numbers.

22. g( f (x)) 5 g(2x 1 9) 5 3(2x 1 9) 2 1 5 6x 1 27 2 1

5 6x 1 26

The functions f and g each have the same domain: all real numbers. So, the domain of g( f (x)) also consists of all real numbers.

23. f ( f (x)) 5 f (2x 1 9) 5 2(2x 1 9) 1 9 5 4x 1 18 1 9

5 4x 1 27

The domain of f consists of all real numbers. So, the domain of f ( f (x)) also consists of all real numbers.

24. g(g(x)) 5 g(3x 2 1) 5 3(3x 2 1) 2 1 5 9x 2 3 2 1

5 9x 2 4

The domain of g consists of all real numbers. So, the domain of g (g(x)) also consists of all real numbers.

25. y 5 22x 1 5

x 5 22y 1 5

x 2 5 5 22y

2 1 } 2 x 1

5 } 2 5 y

26. y 5 1 } 3 x 1 4

x 5 1 } 3 y 1 4

x 2 4 5 1 } 3 y

3x 2 12 5 y

27. f (x) 5 5x 2 12

y 5 5x 2 12

x 5 5y 2 12

x 1 12 5 5y

1 } 5 x 1

12 } 5 5 y

1 } 5 x 1

12 } 5 5 f 21(x)


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409Algebra 2


28. y 5 1 } 2 x4, x ≥ 0

x 5 1 } 2 y4

2x 5 y4

4 Î}

2x 5 y

29. f (x) 5 x3 1 5

y 5 x3 1 5

x 5 y3 1 5

x 2 5 5 y3

3 Î}

x 2 5 5 y

3 Î}

x 2 5 5 f 21(x)

30. f (x) 5 22x3 1 1

y 5 22x3 1 1

x 5 22y3 1 1

x 2 1 5 22y3

1 2 x

} 2 5 y3

3 Î}

1 2 x

} 2 5 y

3 Î} (1 2 x)4

} 2 p 4 5 y

3 Î}

4 2 4x }

3 Î}

8 5 y

3

Î}

24x 1 4 }

2 5 y

3 Î}

24x 1 4 }

2 5 f 21(x)

31. y 5 26 3 Î}

x

x 22 21 0 1 2

y 7.56 6 0 26 27.56

8

x

y

22


32. y 5 Ï}

x 2 4 2 2


x to the right 4 units and down 2 units.

1

x

y

22


33. f (x) 5 2 3 Ï}

x 1 3 1 4

Because h 5 23 and k 5 4, shift the graph of f (x) 5 2

3 Ï}

x to the left 3 units and up 4 units.

x

y

21

1


34. Ï}

3x 1 7 5 4

( Ï}

3x 1 7 )2 5 42

3x 1 7 5 16

3x 5 9

x 5 3

Check: Ï}

3(3) 1 7 0 4

Ï}

16 0 4

4 5 4 ✓

35. Ï}

3x 2 Ï}

x 1 6 5 0

Ï}

3x 5 Ï}

x 1 6

( Ï}

3x )2 5 ( Ï

}

x 1 6 )2

3x 5 x 1 6

2x 5 6

x 5 3

Check: Ï}

3(3) 2 Ï}

3 1 6 0 0

Ï}

9 2 Ï}

9 0 0

0 5 0

36. x 2 3 5 Ï}

x 2 1

(x 2 3)2 5 ( Ï}

x 2 1 )2

x2 2 6x 1 9 5 x 2 1

x2 2 7x 1 10 5 0

(x 2 5)(x 2 2) 5 0

x 2 5 5 0 or x 2 2 5 0

x 5 5 or x 5 2


5 2 3 0 Ï}

5 2 1 2 2 3 0 Ï}

2 2 1

2 0 Ï}

4 21 0 Ï}

1

2 5 2 ✓ 21 Þ 1

The solution is x 5 5.

37. a. E 5 625s2

E }

625 5 s2

Ï}

E }

625 5 s

Ï

} E }

25 5 s


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b. When E 5 120,000:

s 5 Ï

} E } 25 5

Ï}

120,000 } 25 ø 13.9 meters per second

13.9 m

} sec

p 60 sec }

1 min p 60 min

} 1 h

p 1 km }

1000 m ø 50

km }

h

The speed of the car is about 50 kilometers per hour.

c. No; doubling the kinetic energy will increase the speed by a factor of Ï

}

2 .

38. h 5 0.9(200 2 a)

h 5 180 2 0.9a

0.9a 5 180 2 h

a 5 200 2 h } 0.9

a 5 200 2 36

} 0.9 5 160

Your average is 160 if your handicap is 36.

Standardized Test Preparation (p. 471)

1. Partial credit

For part a, the composition is formed and simplifi ed correctly.

For part b, the steps are correct for fi nding h(l(20)), but the last computation is incorrect.

For part c, the explanation is clear; however, the height at the hip is incorrect from part b.

2. No credit

For part a, the composition is formed incorrectly.

For part b, the student substitutes 10 for f rather than 20.

For part c, the student’s description is incorrect.

Standardized Test Practice (pp. 472–473)

1. a. w 5 24

} 32 p b. d 5 220w

w 5 3 } 4 p d 5 220 1 3 }

4 p 2

d 5 165p

c. 5th gear: w 5 24

} 19 p → d 5 220 1 24 }

19 2 p ø 278p

10th gear: w 5 40

} 22 p → d 5 220 1 40 }

22 2 p ø 400 p

15th gear: w 5 50

} 19 p → d 5 220 1 50 }

19 2 p ø 579 p

Gear 5th 10th 15th

Teeth inChainwheel 24 40 50

Teeth inFreewheel 19 22 19

Distance perpedal revolution 278 in. 400 in. 579 in.

As the gear increases, the distance traveled per pedal revolution increases.

2. a. 2 people: s 5 4.62 9 Î}

2 ø 5 4 people: s 5 4.62 9

Î}

4 ø 5.4

8 people: s 5 4.62 9 Î}

8 ø 5.8

The speeds of a boat for crews of 2 people, 4 people, and 8 people are about 5 meters per second, 5.4 meters per second, and 5.8 meters per second respectively.

b. distance 5 rate p time

2 people : 2000 5 5t

400 5 t

400 seconds p 1 minute }

60 seconds ø 6.7 minutes

4 people : 2000 5 5.4t

370 5 t

370 seconds p 1 minute }


8 people : 2000 5 5.8t

344.8 ø t

344.8 seconds p 1 minute }


c. No, doubling the number of rowers does not double the speed; it increased the speed by about 0.4 meter per second.

3. a. V 5 s3 5 43 5 64

The volume of the cube is 64 cubic inches.

b. Triangle base: Pentagon base

V 5 0.433ls2 V 5 1.72ls2

64 5 0.433s3 64 5 1.72s3

148 ø s3 37 ø s3

3 Î}

148 ø s 3 Î}

37 ø s 5.3 ø s 3.3 ø s The side length The side length is about 5.3 inches. is about 3.3 inches.

Hexagon base: Octagon base:

V 5 2.60ls2 V 5 4.83ls2

64 5 2.60s3 64 5 4.83s3

25 ø s3 13 ø s3

3 Î}

25 ø s 3 Î}

13 ø s3

2.9 ø s 2.4 ø s The side length The side length

is about 2.9 inches. is about 2.4 inches.

Decagon base:

V 5 7.69ls2

64 5 7.69s3

8 ø s3

3 Î}

8 ø s 2 ø s The side length is about 2 inches.

c. No; from part (b) the volume was the same for all of the prisms when the prisms had different side lengths.


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411Algebra 2


4. a. S 5 3πr2

S }

3π� 5 r2

Ï}

S }

3π� 5 r

b. V 5 2 } 3 πr3 5

2 } 3 π 1 Ï

}

S }

3π� 2 3

5 2 π

} 3 1 S } 3π� 2

3/2 5

2 π } 3 Ï}

1 S } 3π� 2

3

5 2πS

} 3 p 3π� Ï}

S }

3π� 5 2S

} 9 Ï}

S }

3π�

c. Volume of smaller snow globe:

V 5 2S

} 9 Ï}

S }

3π�

Volume of the larger snow globe:

V 5 2(2S)

} 9 Ï}

2S

} 3π� 5 2 1 2S

} 9 2 p Ï

}

2 Ï}

S }

Ï}

3π� 5 2 Ï

}

2 1 2 } 9 S Î}

S }

3π� 2 Doubling the surface area causes the volume to increase

by a factor of 2 Ï}

2 .

5. D;

Ï}

2x 1 3 5 5

( Ï}

2x 1 3 )2 5 52

2x 1 3 5 25

2x 5 22

x 5 11

6. C;

Because the graph of y 5 3 Î}

x is shifted up 3 units, h 5 0 and k 5 3.

7. A;

y 5 22x5 1 10

x 5 22y5 1 10

x 2 10 5 22y5

5 2 x } 2 5 y5

5 Î} 5 2 x } 2 5 y

8. (8x)3/5 5 8

[(8x)3/5]5/3 5 85/3

8x 5 32

x 5 4

9. f (x) 5 2x1/2, g(x) 5 3x2

g( f (x)) 5 g(2x1/2) 5 3(2x1/2)2 5 3(4x) 5 12x

g( f (9)) 5 12(9) 5 108

10. 1 55 }

25 2 21/5

5 1 25

} 55 2

1/5 5

(25)1/5

} (55)1/5

5 2 } 5

11. 4 Î} 16x3y4z 5 2xayzb

2y 4 Ï}

x3z 5 2xayzb

x3/4z1/4 5 xazb

a 5 3 } 4 , b 5

1 } 4

a 1 b 5 1

12. 1 }

( 4 Ï}

625 )22 5 ( 4

Î}

625 )2 5 52 5 25

13. f (x) p g(x) 5 2 } 5 x20.25 p 5x3.25 5 2x20.25 1 3.25 5 2x3

When x 5 3:

f (3) p g(3) 5 2(3)3 5 2 p 27 5 54

14. y 5 3 Î}

x 2 8 1 5

y 5 3 Î}

0 2 8 1 5

y 5 3 Î}

28 1 5

y 5 22 1 5

y 5 3

The y-intercept is 3.

15. f (x) 5 g(x)

Ï}

x 2 2 1 3 5 Ï}

x 1 13

( Ï}

x 2 2 1 3)2 5 ( Ï

}

x 1 13 )2

x 2 2 1 6 Ï}

x 2 2 1 9 5 x 1 13

6 Ï}

x 2 2 5 6

Ï}

x 2 2 5 1

( Ï}

x 2 2 )25 12

x 2 2 5 1

x 5 3

16. y 5 3x 1 9

x 5 3y 1 9

x 2 9 5 3y

1 }

3 x 2 3 5 y

The slope of the graph of the inverse is 1 }

3 .

17. 2 Ï}

x 1 4 5 Ï}

x 2 1 1 1

(2 Ï}

x 1 4 )2 5 ( Ï

}

x 2 1 1 1)2

x 1 4 5 x 2 1 1 2 Ï}

x 2 1 1 1

4 5 2 Ï}

x 2 1

2 5 Ï}

x 2 1

22 5 ( Ï}

x 2 1 )2

4 5 x 2 1

5 5 x

Check: 2

Î}

5 1 4 0 Ï}

5 2 1 1 1

2 Ï}

9 0 Ï}

4 1 1

23 Þ 3

There is no solution because 5 is extraneous. The results are the same because both methods yield no solution.


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18. y 5 0.03937x

y }

0.03937 5 x

The inverse function is y 5 25.4x.

From the inverse function, you can convert x inches to y millimeters.

3 inches: w 5 25.4(3) ø 76

5 inches: l 5 25.4(5) ø 127

A 5 lw ø 127(76) ø 9652

The area of a 3 inch by 5 inch index card is about 9652 square millimeters.

19. Let x be the original price.

Function for $10 gift card: f (x) 5 x 2 10

Function for 15% discount: g (x) 5 x 2 0.15x 5 0.85x

g( f (x)) 5 fi nal price when $10 is subtracted before 15% discount is applied.

g( f (x)) 5 g(x 2 10) 5 0.85(x 2 10)

f ( g (x)) 5 the fi nal price when 15% discount is applied before the $10 is subtracted.

f ( g (x)) 5 f (0.85x) 5 0.85x 2 10

f (g(27)) 2 0.85(27 2 10) 5 0.85(17) 5 $14.45

f (g(27)) 5 0.85(27) 2 10 5 22.95 2 10 5 $12.95

$14.45 2 $12.95 5 $1.50

You can save $1.50 if the store applies the 15% discount fi rst.

20. No, f and g are not inverse functions because they are not refl ections in the line y 5 x.

Cumulative Review, Chs. 1–6 (pp. 474–475)

1. (3, 1), m 5 4

y 2 y1 5 m(x 2 x1)

y 2 1 5 4(x 2 3)

y 2 1 5 4x 2 12

y 5 4x 2 11

2. (4, 6), m 5 7

y 2 y1 5 m(x 2 x1)

y 2 6 5 7(x 2 4)

y 2 6 5 7x 2 28

y 5 7x 2 22

3. (23, 2), m 5 28

y 2 y1 5 m(x 2 x1)

y 2 2 5 28(x 2 (23))

y 2 2 5 28(x 1 3)

y 5 28x 2 22

4. (1,25), m 5 9

y 2 y1 5 m(x 2 x1)

y 2 (25) 5 9(x 2 1)

y 1 5 5 9x 2 9

y 5 9x 2 14

5. (25, 8), m 5 4 } 5

y 2 y1 5 m(x 2 x1)

y 2 8 5 4 } 5 F x 2 (25) G

y 2 8 5 4 } 5 (x 1 5)

y 2 8 5 4 } 5 x 1 4

y 5 4 } 5 x 1 12

6. (2, 210), m 5 2 3 } 4

y 2 y1 5 m(x 2 x1)

y 2 (210) 5 2 3 } 4 (x 2 2)

y 1 10 5 2 3 } 4 x 1

3 } 2

y 5 2 3 } 4 x 2

17 } 2

7. 22x 1 7 5 15

22x 5 8

x 5 24

Check: 22(24) 1 7 0 15

8 1 7 0 15

15 5 15 ✓

8. 4x 2 6 5 14

4x 2 6 5 14 or 4x 2 6 5 214

4x 5 20 or 4x 5 28

x 5 5 or x 5 22


4(5) 2 6 0 14 4(22)26 0 14

20 2 6 0 14 28 2 6 0 14

14 5 14 ✓ 14 5 14 ✓

9. x2 2 9x 1 14 5 0

(x 2 7)(x 2 2) 5 0

x 2 7 5 0 or x 2 2 5 0

x 5 7 or x 5 2

Check x 5 7: or Check x 5 2:

72 2 9(7) 1 14 0 0 22 2 9(2) 1 14 0 0

49 2 63 1 14 0 0 4 2 18 1 14 0 0

214 1 14 0 0 214 1 14 0 0

0 5 0 ✓ 0 5 0 ✓

10. 4x2 2 6x 1 9 5 0

x 5 2(26) 6 Ï

}}

(26)2 2 4(4)(9) }}}

2(4)

x 5 6 6 Ï

}

2108 }

28

x 5 3 6 3i Ï

}

3 } 4


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413Algebra 2


Check 3 2 3i Ï

}

3 }

4 :

4 1 3 2 3i Ï}

3 }

4 2

2

2 6 1 3 2 3i Ï}

3 }

4 2 1 9 0 0

4 1 9 2 18i Ï}

3 1 27i2 }}

16 2 2

18 2 18i Ï}

3 } 4 1 9 0 0

9 2 18i Ï

}

3 2 27 2 18 1 18i Ï}

3 }}}

4 1 9 0 0

2 36

} 4 1 9 0 0

0 5 0 ✓

Check 3 1 3i Ï

}

3 }

4 :

4 1 3 1 3i Ï}

3 }

4 2

2

26 1 3 1 3i Ï}

3 }

4 2 1 9 0 0

4 1 9 1 18i Ï}

3 1 27i2 }}

16 2 2

18 1 18i Ï}

3 } 4 1 9 0 0

9 1 18i Ï

}

3 2 27 2 18 2 18i Ï}

3 }}}

4 1 9 0 0

2 36

} 4 1 9 0 0

0 5 0 ✓

11. x3 1 3x2 2 10x 5 0

x(x2 1 3x 2 10) 5 0

x(x 1 5)(x 2 2) 5 0

x 5 0, x 5 25, or x 5 2

Check x 5 0:

03 1 3(0)2 2 10(0) 0 0

0 5 0 ✓

Check x 5 25:

(25)3 1 3(25)2 2 10(25) 0 0

2125 1 75 1 50 0 0

250 1 50 0 0

0 5 0 ✓

Check x 5 2:

23 1 3(22) 2 10(2) 0 0

8 1 12 2 20 0 0

20 2 20 0 0

0 5 0 ✓

12. Ï}

8x 1 1 5 7

( Ï}

8x 1 1 )2 5 72

8x 1 1 5 49

8x 5 48

x 5 6

Check: Ï}

8(6) 1 1 0 7

Ï}

49 0 7

7 0 7 ✓

13. 1

x

y

21

14.

x

y

21

1

15.

1

x

y

21

16.

1

x

y

22

17.

1

x

y

1

18.

1

x

y

21

19. y 5 x3 2 2

x 23 22 21 0 1 2 3

y 229 210 23 22 21 6 25

2

x

y

21

20. y 5 3(x 1 2)(x 2 1)2

x 24 23 21 0 2 3

y 2150 248 12 6 12 60

2

x

y

21

21.

1

x

y

1


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22. 2x 1 5y 5 1 3 (23) 26x 2 15y 5 23

3x 2 2y 5 30 3 2 6x 2 4y 5 60

219y 5 57

y 5 23

2x 1 5(23) 5 1

2x 2 15 5 1

2x 5 16

x 5 8


23. 3x 2 y 5 29 3 3 9x 2 3y 5 227

4x 1 3y 5 14 4x 1 3y 5 14

13x 5 213

x 5 21

3(21) 2 y 5 29

23 2 y 5 29

2y 5 26

y 5 6


24. 2x 1 3y 5 47

7x 2 8y 5 22

2x 2 y 1 3z 5 219

2x 1 3y 5 47 3 8 16x 1 24y 5 376

7x 2 8y 5 22 3 3 21x 2 24y 5 26

37x 5 370

x 5 10

2(10) 1 3y 5 47

20 1 3y 5 47

3y 5 27

y 5 9

2(10) 2 9 1 3z 5 219

20 2 9 1 3z 5 219

3z 5 230

z 5 210

The solution is x 5 10, y 5 9 and z 5 210, or the ordered triple (10, 9, 210).

25. (4 2 2i) 1 (5 1 i) 5 (4 1 5) 1 (22 1 1)i 5 9 2 i

26. (3 1 4i) 2 (7 1 2i) 5 (3 2 7) 1 (4 2 2)i 5 24 1 2i

27. (4 2 2i)(6 1 5i) 5 24 1 20i 2 12i 2 10i2

5 24 1 8i 2 10(21)

5 34 1 8i

28. y 5 x2 1 6x 1 16

y 1 9 5 (x2 1 6x 1 9) 1 16

y 1 9 5 (x 1 3)2 1 16

y 5 (x 1 3)2 1 7

29. y 5 2x2 1 12x 2 46

y 2 36 5 2(x2 2 12x 1 36) 2 46

y 2 36 5 2(x 2 6)2 2 46

y 5 2(x 2 6)2 2 10

30. y 5 2x2 2 4x 1 7

y 1 2(1) 5 2(x2 2 2x 1 1) 1 7

y 1 2 5 2(x 2 1)2 1 7

y 5 2(x 2 1)2 1 5

31. (2x3y2)3 5 23x9y6 5 8x9y6

32. (x8)23/4 5 x26 5 1 }

x6

33. x3y24

} x24y25 5 x3 2 (24)y24 2 (25) 5 x7y

34. 1 x2y1/3

} x1/4y

2 2

5 (x2 2 1/4y1/3 2 1)2

5 (x7/4y22/3)2

5 x7/2y24/3

5 x 7/2

} y 4/3

35. (x2 1 11x 2 9) 1 (4x2 2 5x 2 7) 5 x2 1 4x2 1 11x 2 5x 2 9 2 7

5 5x2 1 6x 2 16

36. (x3 1 3x 2 10) 2 (2x3 1 3x2 1 8x) 5 x3 1 3x 2 10 2 2x3 2 3x2 2 8x

5 2x3 2 3x2 2 5x 2 10

37. (2x 2 5)(x2 1 4x 2 7) 5 (2x 2 5)x2 1 (2x 2 5)4x 1 (2x 2 5)(27)

5 2x3 2 5x2 1 8x2 2 20x 2 14x 1 35

5 2x3 1 3x2 2 34x 1 35

38. (x3 2 10x2 1 33x 2 28) 4 (x 2 5)

5 1 210 33 228

5 225 40

1 25 8 12

x3 2 10x2 1 33x 2 28

}} x 2 5 5 x2 2 5x 1 8 1 12 } x 2 5

39. x4 2 3x2 2 40 5 (x2 2 8)(x2 1 5) 40. x3 2 125 5 x3 2 53 5 (x 2 5)(x2 1 5x 1 25) 41. x3 2 6x2 2 9x 1 54 5 x2(x 2 6) 2 9(x 2 6)

5 (x2 2 9)(x 2 6) 5 (x 1 3)(x 2 3)(x 2 6)

42. f (x) 1 g (x) 5 (2x 2 6) 1 (5x 1 1) 5 7x 2 5



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415Algebra 2


43. f (x) p g(x) 5 (2x 2 6)(5x 1 1) 5 10x2 2 28x 2 6

The functions f and g each have the same domain: all real numbers. So, the domain of f p g also consists of all real numbers.

44. f ( g (x)) 5 f (5x 1 1) 5 2(5x 1 1) 2 6 5 10x 1 2 2 6

5 10x 2 4

The functions f and g each have the same domain: all real numbers. So, the domain of f ( g (x)) also consists of all real numbers.

45. g( f (x)) 5 g(2x 2 6) 5 5(2x 2 6) 1 1 5 10x 2 30 1 1

5 10x 2 29

The functions f and g each have the same domain: all real numbers. So, the domain of g( f (x)) also consists of all real numbers.

46. f (x) 5 4x 1 6

y 5 4x 1 6

x 5 4y 1 6

x 2 6 5 4y

1 }

4 x 2

3 } 2 5 y

f 21(x) 5 1 } 4 x 2

3 } 2

47. f (x) 5 3 } 7 x 1 7

y 5 3 } 7 x 1 7

x 5 3 } 7 y 1 7

x 2 7 5 3 } 7 y

7 }

3 x 2

49 } 3 5 y

f 21(x) 5 7 } 3 x 2

49 } 3

48. f (x) 5 1 } 3 x 2

2 } 3

y 5 1 } 3 x 2

2 } 3

x 5 1 } 3 y 2

2 } 3

x 1 2 } 3 5

1 } 3 y

3x 1 2 5 y

f 21(x) 5 3x 1 2

49. f (x) 5 x3 2 5

} 6

y 5 x3 2 5

} 6

x 5 y3 2 5

} 6

6x 5 y3 2 5

6x 1 5 5 y3

3 Î}

6x 1 5 5 y

f 21(x) 5 3 Î}

6x 1 5

50. f (x) 5 3 Î} 2x 1 7

} 3

y 5 3 Î} 2x 1 7

} 3

x 5 3 Î} 2y 1 7

} 3

x3 5 2y 1 7

} 3

3x3 5 2y 1 7

3x3 2 7 5 2y

3x3 2 7

} 2 5 y

f 21(x) 5 3x3 2 7

} 2

51. f (x) 5 2 8 } 9 x 5 1 2

y 5 2 8 } 9 x 5 1 2

x 5 2 8 } 9 y 5 1 2

x 2 2 5 2 8 } 9 y 5

2 9 } 8 x 1

9 } 4 5 y 5

5

Î} 2 9 } 8 x 1

9 } 4 5 y

f 21(x) 5 5 Î} 2

9 } 8 x 1

9 } 4

52. Cost of bicycle

5 Amount savedper week

p Number of weeks

360 5 30 p x

360 5 30x

12 5 x

It will take 12 weeks to save enough money to buy the bicycle.


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53.

0 2 41 3 5 6 7 80

1.01.52.02.53.03.54.04.55.05.5

Years of existence

Do

nat

ion

s(m

illio

ns

of

do

llars

)

x

y

Charitable Donations

(x1, y1) 5 (2, 2.3); (x2, y2) 5 (6, 4.3)

m 5 4.3 2 2.3

} 6 2 2 5 2 } 4 5

1 } 2

y 2 y1 5 m(x 2 x1)

y 2 2.3 5 1 } 2 (x 2 2)

y 2 2.3 5 1 } 2 x 2 1

y 5 1 } 2 x 1 1.3

When x 5 8: y 5 1 } 2 (8) 1 1.3 5 5.3

The organization will receive about 5.3 million dollars in the eighth year of its existence.

54. x 5 Number of lower-level tickets

y 5 Number of upper-level tickets

x 1 y 5 9800 3 (225) 225x 2 25y 5 224,500

35x 1 25y 5 280,000 35x 1 25y 5 280,000

10x 5 35,000

x 5 3500

3500 1 y 5 9800

y 5 6300

There are 3500 lower-level seats and 6300 upper-level seats sold for the ice show.

55. Ticket Cost ($)

Student F 4

8

6G

Adult

Senior Citizen

Friday Night Attendance

Student F 140

170 55G

Adult

Senior Citizen

Saturday Night Attendance

Student F 126

188

64G

Adult

Senior Citizen

1 F 140 170 55 G 1 F 126 188 64 G 2 F 4

8

6G

5

F 266 358 119 G F 4

8

6G

5 F 4642 G

The income for ticket sales for Friday and Saturday nights’ concerts was $4642.

56. h 5 216t2 1 h0 → h 5 216t2 1 85

When h 5 0: 0 5 216t2 1 85

285 5 216t2

5.3 ø t2

6 Ï}

5.3 ø t 62.3 ø t Reject the negative solution, 22.3. The rock will hit the

ground in about 2.3 seconds.

57. (0,6), (20,56), (36,24)

y 5 ax2 1 bx 1 c

6 5 a(0)2 1 b(0) 1 c → 6 5 c

56 5 a(20)2 1 b(20) 1 c → 56 5 400a 1 20b 1 c

24 5 a(36)2 1 b(36) 1 c → 24 5 1296a 1 36b 1 c

400a 1 20b 1 6 5 56 → 400a 1 20b 5 50

1296a 1 36b 1 6 5 24 → 1296a 1 36b 5 18

400a 1 20b 5 50 3 9 3600a 1 180b 5 450

1296a 1 36b 5 18 3 (25) 26480a 2 180b 5 290

22880a 5 360

a 5 2 1 } 8

400 1 2 1 } 8 2 1 20b 5 50

250 1 20b 5 50

20b 5 100

b 5 5

The solution is a 5 2 1 } 8 , b 5 5, and c 5 6. A quadratic

function that models the baseball’s path is

y 5 2 1 } 8 x2 1 5x 1 6.


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417Algebra 2


58. Volume 5 lwh

200 5 (x)(x)(x 1 3)

200 5 x2(x 1 3)

0 5 x3 1 3x2 2 200

Possible rational zeros: 62, 64, 65, 68, 610, 620, 625, 640, 650, 6100, 6200.

5 1 3 0 2200

5 40 200

1 8 40 0

x 2 5 5 0 or x2 1 8x 1 40 5 0

x 5 5 No real solution

x3 1 3x2 2 200 5 (x 2 5)(x2 1 8x 1 40) The dimensions of a mold are 5 inches by 5 inches

by 8 inches.

59. 2 4 18 50 106 192

2 14 32 56 86

12 18 24 30

6 6 6

First-orderdifferencesSecond-orderdifferencesThird-orderdifferences

The third-order differences are constant, so the data can be represented by a cubic function. Using the cubic regression feature on a graphing calculator, a model is y 5 x3 2 5x 1 6. When x 5 9: y 5 93 2 5(9) 1 6 5 690

The profi t in the ninth month will be $690.

60. V 5 4 } 3 πr3

7240 5 4 } 3 πr3

5430 5 πr3

5430

} π 5 r3

3 Î}

5430

} π 5 r

12 ø r The radius of the beach ball is about 12 inches.


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