1. Chapter 8

3. Determine whether each expression is a monomial. Explain your reasoning.xy d. c. b. a.
Reason
Monomial?
Expression no
The expression involves subtraction, not the product, of two variables.
yes
The expression is the product of a number and two variables.
yes
is a real number and an example of a constant.
yes
The expression is the product of two variables.
Example 1-1a

4. Determine whether each expression is a monomial. Explain your reasoning.b. a.
Reason
Monomial?
Expression
yes
Single variables are monomials. no
The expression involves subtraction, not the product, of two variables. no
The expression is the quotient, not the product, of two variables.
yes
The expression is the product of a number, , and two variables.
Example 1-1b

5. Lesson 8-1 and 8-2: Multiplying and Dividing Powers

7. 5 x 5 = 25
52 = 81
34 =
3 x 3 x 3 x 3 =
343
7 x 7 x 7 =
73 =

8. Multiplying Powers 22 • 22 = 22+2= 24 = 16 x9 • x = x9+1 =x10
Rule #1: When multiplying powers with the same base, ADD the exponents.
22 • 22 = 22+2= 24 = 16
x9 • x = x9+1 =x10
36 • 3-2 = = 34 = 81

9. Simplify. 1

10. Simplifying Variable Expressions

11. Communicative and Associative Properties
Simplify .
Communicative and Associative Properties
Product of Powers
Simplify.
Answer:
Example 1-2b

12. Simplify each expression. a.b.
Answer:
Answer:
Example 1-2c

13. Simplify Power of a Power Simplify. Power of a Power Simplify. Answer:
Example 1-3a

14. Simplify
Answer:
Example 1-3b

15. Simplify Power of a Power Power of a Product Power of a Power
Example 1-5a

16. Commutative Property
Answer:
Power of Powers
Example 1-5b

17. Simplify
Answer:
Example 1-5c

18. End of Lesson 1

19. Add or Multiply ???

20. DIVIDING MONOMIALS 68 = 68-5 =63 65
To divide powers with the same base, subtract the exponent of the denominator from the exponent of the numerator.
68 = 68-5 =63 65

21. Quotient of Powers Property
Dividing Powers
Quotient of Powers Property
Rule #2: When dividing powers with the same base, SUBTRACT the exponents.

22. Simplify.

24. Simplify Assume that x and y are not equal to zero.
Group powers that have the same base.
Quotient of Powers
Answer:
Simplify.
Example 2-1a

25. Simplify Assume that a and b are not equal to zero.
Answer:
Example 2-1b

27. Simplify Assume that e and f are not equal to zero.
Power of a Quotient
Power of a Product
Power of a Power
Answer:
Example 2-2a

28. Example 2-2b

29. Zero Exponents 40 = For any nonzero number a, a0 = 1
Anything to the zero power equals 1 (except zero)
40 =
1000 = 1 1

30. Negative Exponents
For any nonzero number a and any integer n,
a-n = 1/an
5-2 = 1 52

31. 1
3-5 =
3y-2 = 3 y2 35 1
5-2 =
a-7b3 = b3 52 a7

32. =
5-11
5-8 x 5-3 = 1 or
511
a = a8
a-2 x a10 =

33. b =
b-3
b-8 x b5 = 1 or b3 = 37
3-4 x 311 =

34. 35 3-3 35 - 8 38 a6 a8 a6 – (-2) a-2 m2 m2 – (-4) m6 m-4 33 1 = or = =

35. Simplify Assume that m and n are not equal to zero.
Answer: 1
Example 2-3a

36. Simplify . Assume that m and n are not equal to zero.
Answer:
Quotient of Powers
Example 2-3b

37. Simplify each expression. Assume that z is not equal to zero.b.
Answer: 1
Answer:
Example 2-3c

38. Simplify . Assume that y and z are not equal to zero.
Write as a product of fractions.
Answer:
Multiply fractions.
Example 2-4a

39. Simplify . Assume that p, q, and r are not equal to zero.
Group powers with the same base.
Quotient of Powers and Negative Exponent Properties
Example 2-4b

40. Negative Exponent Property
Simplify.
Negative Exponent Property
Multiply fractions.
Answer:
Example 2-4c

41. Simplify each expression. Assume that no denominator is equal to zero.b.
Answer:
Answer:
Example 2-4d

42. End of Lesson 2

44. Scientific Notation
is a short hand way of writing numbers using powers of 10

45. Standard Product Scientific Notation Form Notation
120,000, x 108
1.2 x 100,000,000

46. Write in scientific notation.
4.62 x 109
46,200,000,000 =
Where is the decimal now?
Move the decimal to the right of the first significant digit.

47. Write in scientific notation.
8.9 x 107
89,000,000 =
Where is the decimal now?
Move the decimal to the right of the first significant digit.

48. Write in scientific notation.
3.04 x 1011
304,000,000,000 =
Where is the decimal now?
Move the decimal to the right of the first significant digit.

49. Standard Product Scientific Notation Form Notation
x 10-4
5.6 x

50. Express in standard notation.
move decimal point 3 places to the left.
Answer:
Example 3-1a

51. Express in standard notation.
move decimal point 5 places to the right.
Answer: 219,000
Example 3-1b

52. Express each number in standard notation. a.
Answer:
Answer:
Example 3-1c

53. Express 0.000000672 in scientific notation.
Move decimal point 7 places to the right.
Answer:
Example 3-2a

54. Express 3,022,000,000,000 in scientific notation.
The decimal point moved 12 places to the left.
Answer:
Example 3-2b

55. Express each number in scientific notation.
Answer:
Answer:
Example 3-2c

56. Evaluate Express the result in scientific and standard notation.
Commutative and Associative Properties
Product of Powers
Associative Property
Product of Powers
Answer:

57. Evaluate Express the result in scientific and standard notation.
Answer:
Example 3-4c

58. Evaluate Express the result in scientific and standard notation.
Associative Property
Product of Powers
Answer:
Example 3-5a

59. Evaluate Express the result in scientific and standard notation.
Answer:
Example 3-5b

60. End of Lesson 3

62. Monomial, Binomial, or Trinomial
State whether each expression is a polynomial. If it is a polynomial, identify it as a monomial, binomial, or trinomial.
Monomial, Binomial, or Trinomial
Polynomial?
Expression a. b. c. d.
Yes, is the difference of two real numbers.
binomial
Yes, is the sum and
difference of three monomials.
trinomial
No are not monomials.
none of these
monomial
Yes, has one term.

63. Monomial, Binomial, or Trinomial
State whether each expression is a polynomial. If it is a polynomial, identify it as a monomial, binomial, or trinomial.
Monomial, Binomial, or Trinomial
Polynomial?
Expression a. b. c. d.
Yes, Is the sum of three monomials.
trinomial
No which is not a monomial.
none of these
Yes, The expression is the sum of two monomials.
binomial
Yes, has one term.
monomial
Example 4-1b

64. Find the degree of each polynomial.b. a.
Degree of Polynomial
Degree of Each Term
Terms
Polynomial
0, 1, 2, 3 3
2, 1, 0 2
1,2,5,1 5
Example 4-3a

65. Find the degree of each polynomial.b. a.
Degree of Polynomial
Degree of Each Term
Terms
Polynomial
2, 1, 3, 0 3
2, 4, 3 4
7, 6 7
Example 4-3b

66. so that the powers of x are in ascending order.
Arrange the terms of
so that the powers of x are in ascending order.
Answer:
Example 4-4a

67. Arrange the terms of so that the powers of x are in ascending order.
Answer:
Example 4-4b

68. Arrange the terms of each polynomial so that the powers of x are in ascending order.b.
Answer:
Answer:
Example 4-4c

69. Arrange the terms of so that the powers of x are in descending order.
Answer:
Example 4-5a

70. Arrange the terms of so that the powers of x are in descending order.
Answer:
Example 4-5b

71. Arrange the terms of each polynomial so that the powers of x are in descending order.b.
Answer:
Answer:
Example 4-5c

72. End of Lesson 4

73. Group like terms together.
Find
Method 1 Horizontal
Group like terms together.
Associative and Commutative Properties
Add like terms.
Example 5-1a

74. Method 2 Vertical
Align the like terms in columns and add.
Notice that terms are in descending order with like terms aligned.
Answer:

75. Find
Answer:
Example 5-1c

76. Subtract by adding its additive inverse.
Find
Method 1 Horizontal
Subtract by adding its additive inverse.
The additive inverse of is
Group like terms.
Add like terms.
Example 5-2a

77. Method 2 Vertical
Align like terms in columns and subtract by adding the additive inverse.
Add the opposite.
Answer: or
Example 5-2b

78. Find
Answer:
Example 5-2c

79. End of Lesson 5

80. Distributive Property
Find
Method 1 Horizontal
Distributive Property
Multiply.
Example 6-1a

81. Distributive Property
Find
Method 2 Vertical
Distributive Property
Multiply.
Answer:
Example 6-1b

82. Find
Answer:
Example 6-1c

83. Distributive Property
Simplify
Distributive Property
Product of Powers
Commutative and Associative Properties
Combine like terms.
Answer:
Example 6-2a

84. Simplify
Answer:
Example 6-2b

85. Distributive Property
Solve
Original equation
Distributive Property
Combine like terms.
Subtract from each side.
Example 6-4a

86. Add 7 to each side. Add 2b to each side. Divide each side by 14.
Answer:
Example 6-4b

87. Check Original equation Simplify. Multiply. Add and subtract.
Example 6-4c

88. Solve
Answer:
Example 6-4d

89. End of Lesson 6

90. Find
Method 1 Vertical
Multiply by –4.
Example 7-1a

91. Find
Multiply by y.
Example 7-1b

92. Find
Add like terms.
Example 7-1c

93. Distributive Property
Find
Method 2 Horizontal
Distributive Property
Distributive Property
Multiply.
Combine like terms.
Answer:
Example 7-1d

94. Find
Answer:
Example 7-1e

95. Find F L O I
Multiply.
Combine like terms.
Answer:
Example 7-2a

96. Find F I O L
Multiply.
Answer:
Combine like terms.
Example 7-2b

97. Find each product. a. b.
Answer:
Answer:
Example 7-2c

98. Distributive Property
Find
Distributive Property
Distributive Property
Answer:
Combine like terms.
Example 7-4a

99. Distributive Property
Find
Distributive Property
Distributive Property
Answer:
Combine like terms.
Example 7-4b

100. Find each product. a. b.
Answer:
Answer:
Example 7-4c

101. End of Lesson 7

102. Find
Square of a Sum
Answer:
Simplify.
Example 8-1a

103. Check Check your work by using the FOIL method.
Example 8-1b

104. Find
Square of a Sum
Answer:
Simplify.
Example 8-1c

105. Find each product. a. b.
Answer:
Answer:
Example 8-1d

106. Find
Square of a Difference
Answer:
Simplify.
Example 8-2a

107. Find
Square of a Difference
Answer:
Simplify.
Example 8-2b

108. Find each product. a. b.
Answer:
Answer:
Example 8-2c

109. Product of a Sum and a Difference
Find
Product of a Sum and a Difference
Answer:
Simplify.
Example 8-4a

110. Product of a Sum and a Difference
Find
Product of a Sum and a Difference
Answer:
Simplify.
Example 8-4b

111. Find each product. a. b.
Answer:
Answer:
Example 8-4c

112. End of Lesson 8

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