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5.7 Review Exercises and Sample Exam

Review Exercises

Rules of Exponents

Simplify.

1. 7376

2. 5956

3. y5y2y3

4. x3y2xy3

5. 5a3b2c6a2bc2

6. 55x2yz55xyz2

7. (3 a 2 b 42 c 3)2

8. (2 a 3b4 c 4)3

9. 5x3y0(z2)32x4(y3)2z

10. (25x6y5z)0

11. Each side of a square measures 5x2 units. Find the area of the square in terms of x.

12. Each side of a cube measures 2x3 units. Find the volume of the cube in terms of x.

Introduction to Polynomials

Classify the given polynomial as a monomial, binomial, or trinomial and state the degree.

13. 8a31

14. 5y2y+1

15. 12ab2

16. 10

Write the following polynomials in standard form.

17. 7x25x

18. 5x213x+2x3

Evaluate.

19. 2x2x+1, where x=3

20. 12x34, where x=13

21. b24ac, where a=12, b=3, and c=32

22. a2b2, where a=12 and b=13

23. a3b3, where a=2 and b=1

24. xy22x2y, where x=3 and y=1

25. Given f(x)=3x25x+2, find f(2).

26. Given g(x)=x3x2+x1, find g(1).

27. The surface area of a rectangular solid is given by the formula SA=2lw+2wh+2lh, where l, w, and h represent the length, width, and height, respectively. If the length of a rectangular solid measures 2 units, the width measures 3 units, and the height measures 5 units, then calculate the surface area.

28. The surface area of a sphere is given by the formula SA=4πr2, where r represents the radius of the sphere. If a sphere has a radius of 5 units, then calculate the surface area.

Adding and Subtracting Polynomials

Perform the operations.

29. (3x4)+(9x1)

30. (13x19)+(16x+12)

31. (7x2x+9)+(x25x+6)

32. (6x2y5xy23)+(2x2y+3xy2+1)

33. (4y+7)(6y2)+(10y1)

34. (5y23y+1)(8y2+6y11)

35. (7x2y23xy+6)(6x2y2+2xy1)

36. (a3b3)(a3+1)(b31)

37. (x5x3+x1)(x4x2+5)

38. (5x34x2+x3)(5x33)+(4x2x)

39. Subtract 2x1 from 9x+8.

40. Subtract 3x210x2 from 5x2+x5.

41. Given f(x)=3x2x+5 and g(x)=x29, find (f+g)(x).

42. Given f(x)=3x2x+5 and g(x)=x29, find (fg)(x).

43. Given f(x)=3x2x+5 and g(x)=x29, find (f+g)(2).

44. Given f(x)=3x2x+5 and g(x)=x29, find (fg)(2).

Multiplying Polynomials

Multiply.

45. 6x2(5x4)

46. 3ab2(7a2b)

47. 2y(5y12)

48. 3x(3x2x+2)

49. x2y(2x2y5xy2+2)

50. 4ab(a28ab+b2)

51. (x8)(x+5)

52. (2y5)(2y+5)

53. (3x1)2

54. (3x1)3

55. (2x1)(5x23x+1)

56. (x2+3)(x32x1)

57. (5y+7)2

58. (y21)2

59. Find the product of x21 and x2+1.

60. Find the product of 32x2y and 10x30y+2.

61. Given f(x)=7x2 and g(x)=x23x+1, find (fg)(x).

62. Given f(x)=x5 and g(x)=x29, find (fg)(x).

63. Given f(x)=7x2 and g(x)=x23x+1, find (fg)(1).

64. Given f(x)=x5 and g(x)=x29, find (fg)(1).

Dividing Polynomials

Divide.

65. 7y214y+287

66. 12x530x3+6x6x

67. 4a2b16ab24ab4ab

68. 6a624a4+5a23a2

69. (10x219x+6)÷(2x3)

70. (2x35x2+5x6)÷(x2)

71. 10x421x316x2+23x202x5

72. x53x428x3+61x212x+36x6

73. 10x355x2+72x42x7

74. 3x4+19x3+3x216x113x+1

75. 5x4+4x35x2+21x+215x+4

76. x44x4

77. 2x4+10x323x215x+302x23

78. 7x417x3+17x211x+2x22x+1

79. Given f(x)=x34x+1 and g(x)=x1, find (f/g)(x).

80. Given f(x)=x532 and g(x)=x2, find (f/g)(x).

81. Given f(x)=x34x+1 and g(x)=x1, find (f/g)(2).

82. Given f(x)=x532 and g(x)=x2, find (f/g)(0).

Negative Exponents

Simplify.

83. (10)2

84. 102

85. 5x3

86. (5x)3

87. 17y3

88. 3x4y2

89. 2a2b5c8

90. (5x2yz1)2

91. (2x3y0z2)3

92. (10 a 5 b 3 c 25a b 2 c 2)1

93. ( a 2 b 4 c 02 a 4 b 3c)3

The value in dollars of a new laptop computer can be estimated by using the formula V=1200(t+1)1, where t represents the number of years after the purchase.

94. Estimate the value of the laptop when it is 1½ years old.

95. What was the laptop worth new?

Rewrite using scientific notation.

96. 2,030,000,000

97. 0.00000004011

Perform the indicated operations.

98. (5.2×1012)(1.8×103)

99. (9.2×104)(6.3×1022)

100. 4×10168×107

101. 9×10304×1010

102. 5,000,000,000,000 × 0.0000023

103. 0.0003/120,000,000,000,000

Sample Exam

Simplify.

1. 5x3(2x2y)

2. (x2)4x3x

3. (2 x 2 y 3)2x2y

4. a. (5)0; b. 50

Evaluate.

5. 2x2x+5, where x=5

6. a2b2, where a=4 and b=3

Perform the operations.

7. (3x24x+5)+(7x2+9x2)

8. (8x25x+1)(10x2+2x1)

9. (35a12)(23a2+23a29)+(115a518)

10. 2x2(2x33x24x+5)

11. (2x3)(x+5)

12. (x1)3

13. 81x5y2z3x3yz

14. 10x915x5+5x25x2

15. x35x2+7x2x2

16. 6x4x313x22x12x1

Simplify.

17. 23

18. 5x2

19. (2x4y3z)2

20. (2 a 3 b 5 c 2a b 3 c 2)3

21. Subtract 5x2y4xy2+1 from 10x2y6xy2+2.

22. If each side of a cube measures 4x4 units, calculate the volume in terms of x.

23. The height of a projectile in feet is given by the formula h=16t2+96t+10, where t represents time in seconds. Calculate the height of the projectile at 1½ seconds.

24. The cost in dollars of producing custom t-shirts is given by the formula C=120+3.50x, where x represents the number of t-shirts produced. The revenue generated by selling the t-shirts for $6.50 each is given by the formula R=6.50x, where x represents the number of t-shirts sold.

a. Find a formula for the profit. (profit = revenuecost)

b. Use the formula to calculate the profit from producing and selling 150 t-shirts.

25. The total volume of water in earth’s oceans, seas, and bays is estimated to be 4.73×1019 cubic feet. By what factor is the volume of the moon, 7.76×1020 cubic feet, larger than the volume of earth’s oceans? Round to the nearest tenth.

Review Exercises Answers

1: 79

3: y10

5: 30a5b3c3

7: 9a4b84c6

9: 10x7y6z7

11: A=25x4

13: Binomial; degree 3

15: Monomial; degree 3

17: x25x+7

19: 22

21: 6

23: −7

25: f(2)=24

27: 62 square units

29: 12x5

31: 8x26x+15

33: 8y+8

35: x2y25xy+7

37: x5x4x3+x2+x6

39: 7x+9

41: (f+g)(x)=4x2x4

43: (f+g)(2)=14

45: 30x6

47: 10y224y

49: 2x4y25x3y3+2x2y

51: x23x40

53: 9x26x+1

55: 10x311x2+5x1

57: 25y2+70y+49

59: x41

61: (fg)(x)=7x323x2+13x2

63: (fg)(1)=45

65: y22y+4

67: a+4b+1

69: 5x2

71: 5x3+2x23x+4

73: 5x210x+1+32x7

75: x3x+5+15x+4

77: x2+5x10

79: (f/g)(x)=x2+x32x1

81: (f/g)(2)=1

83: 1100

85: 5x3

87: y37

89: 2a2c8b5

91: x98z6

93: 8a6b3c3

95: $1,200

97: 4.011×108

99: 5.796×1019

101: 2.25×1020

103: 2.5×1018

Sample Exam Answers

1: 10x5y

3: 4x2y5

5: 60

7: 4x2+5x+3

9: 23a259

11: 2x2+7x15

13: 27x2y

15: x23x+1

17: 18

19: y64x8z2

21: 5x2y2xy2+1

23: 118 feet

25: 16.4