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4. Contents Lesson 4-1Factors and Monomials Lesson 4-2Powers and Exponents Lesson 4-3Prime Factorization Lesson 4-4Greatest Common Factor (GCF) Lesson 4-5Simplifying Algebraic Fractions Lesson 4-6Multiplying and Dividing Monomials Lesson 4-7Negative Exponents Lesson 4-8Scientific Notation

5. Lesson 1 Contents Example 1Use Divisibility Rules Example 2Use Divisibility Rules to Solve a Problem Example 3Find Factors of a Number Example 4Identify Monomials

6. Example 1-1a Determine whether 435 is divisible by 2, 3, 5, 6, or 10. NumberDivisible? Reason 2 3 5 6 10 no The ones digit is 5 and 5 is not divisible by 2. The ones digit is 5. yesno The ones digit is not 0. no 435 is not divisible by 2, so it cannot be divisible by 6. Answer: So, 435 is divisible by 3 and 5. yes The sum of the digits is or 12 and 12 is divisible by 3.

7. Example 1-1b Determine whether 786 is divisible by 2, 3, 5, 6, or 10. Answer: 786 is divisible by 2, 3, and 6.

8. Example 1-2a Student Elections Sonya is running for student council president. She wants to give out campaign flyers with a pen to each student in the school. She can buy “Vote for Sonya” pens in packages of 5, 6, or 10. If there are 306 students in the school and she wants no pens left over, which size packages should she buy? SizeYes/No Reason 5 6 10 no The ones digit of 306 is not 0 or 5. yes 306 is divisible by 2 and 3, so it is also divisible by 6. Therefore, there would be no pens left over. The ones digit is not 0. no Answer: Sonya should buy pens in packages of 6.

9. Example 1-2b Transportation A class of 72 students is taking a field trip. The transportation department can provide vans that seat 5, 6, or 10 students. If the teacher wants all vans to be the same size and no empty seats, what size vans should be used? Answer: Vans that seat 6 should be used.

10. Example 1-3a List all the factors of 64. Use the divisibility rules to determine whether 64 is divisible by 2, 3, 5, and so on. Then use division to find other factors of 64. Number64 Divisible by Number?Factor Pairs 1 2 3 4 5 6 7 8 ___ no ___ no ___ no ___ no yes

11. Example 1-3b Answer: So, the factors of 64 are 1, 2, 4, 8, 16, 32, and 64.

12. Example 1-3c List all the factors of 96. Answer: 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, and 96

13. Determine whether is a monomial. Example 1-4a Simplify. Answer: This expression is not a monomial because in its simplest form, it involves two terms that are added. Distributive Property

14. Example 1-4b Determine whether is a monomial. Answer:This expression is a monomial because it is the product of a rational number,, and a variable, x.

15. Example 1-4c Determine whether each expression is a monomial. Answer: monomial Answer: not a monomial a. b.

16. End of Lesson 1

17. Lesson 2 Contents Example 1Write Expressions Using Exponents Example 2Use Exponents in Expanded Form Example 3Evaluate Expressions

18. Write using exponents. Example 2-1a Answer:The base is 6. It is a factor 4 times, so the exponent is 4.

19. Example 2-1b Write p using exponents. Answer:The base is p. It is a factor 1 time, so the exponent is 1.

20. Example 2-1c Write (–1)(–1)(–1) using exponents. Answer:The base is – 1. It is a factor 3 times, so the exponent is 3.

21. Write using exponents. Example 2-1d Answer:The base is. It is a factor 2 times, so the exponent is 2.

22. Example 2-1e Write each expression using exponents. Answer:First group the factors with like bases. Then write using exponents.

23. Example 2-1f Answer: Write each expression using exponents. a. b. c. d. e.

24. Example 2-2a Express 235,016 in expanded form. Answer: Step 1Use place value to write the value of each digit in the number. Step 2 Write each place value as a power of 10 using exponents.

25. Example 2-2b Express 24,706 in expanded form. Answer:

26. Example 2-3a Answer: 16 4 is a factor two times. Multiply. Evaluate.

27. Example 2-3b –2 is a factor 3 times. Multiply. Subtract. Answer: Replace r with –2. Evaluate.if

28. Example 2-3c Simplify the expression inside the parentheses. Evaluate (0) 2. Replace x with 2 and y with –2. Simplify. Answer: 0 Evaluate. ifand

29. Example 2-3d Evaluate each expression. Answer: 81 Answer: 84 Answer: –24 a. b. if c.

30. End of Lesson 2

31. Lesson 3 Contents Example 1Identify Numbers as Prime or Composite Example 2Write Prime Factorization Example 3Factor Monomials

32. Example 3-1a Determine whether 31 is prime or composite. Find factors of 31 by listing the whole number pairs whose product is 31. The number 31 has only two factors. Answer: Therefore, 31 is a prime number.

33. Example 3-1b Determine whether 36 is prime or composite. Find factors of 36 by listing the whole number pairs whose product is 36. The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36. Answer: Since the number has more than two factors, it is composite.

34. Example 3-1c Determine whether each number is prime or composite. a. 49 b. 29 Answer: composite Answer: prime

35. Example 3-2a Write the prime factorization of 56. ^ ^ ^ \ \\ The prime factorization is complete because 2 and 7 are prime numbers. Answer: The prime factorization of 56 is.

36. Example 3-2b Write the prime factorization of 72. Answer:

37. Example 3-3a Factor. Answer:

38. Example 3-3b Factor. Answer:

39. Example 3-3c Factor each monomial. Answer: a. b. Answer:

40. End of Lesson 3

41. Lesson 4 Contents Example 1Find the GCF Example 2Find the GCF Example 3Use the GCF to Solve Problems Example 4Find the GCF of Monomials Example 5Factor Expressions

42. Example 4-1a Find the GCF of 16 and 24. Method 1 List the factors. factors of 16: 1, 2, 4, 8, 16 factors of 24: 1, 2, 3, 4, 6, 8, 12, 24 Answer: The greatest common factor of 16 and 24 is 8.

43. Example 4-1a Find the GCF of 16 and 24. Method 2 Use prime factorization. Answer: 8 The GCF is the product of the common prime factors. Again, the GCF of 16 and 24 is 8.

44. Example 4-1c Find the GCF of 18 and 30. Answer: 6

45. Example 4-2a Find the GCF of 28 and 35. First, factor each number completely. The common prime factor is 7. Answer: The GCF of 28 and 35 is 7.

46. Example 4-2b Find the GCF of 12, 48, and 72. First, factor each number completely. The common prime factors are 2, 2, and 3. Answer: The GCF of 12, 48, and 72 is.

47. Example 4-2c Find the GCF of each set of numbers. a. 24, 32 b. 30, 42, 60 Answer: 8 Answer: 6

48. Example 4-3a Cookie Sale Parents donated 150 chocolate chip cookies and 120 molasses cookies for a school bake sale. If the cookies are arranged on plates, and each plate has the same number of chocolate chip cookies and the same number of molasses cookies, what is the largest number of plates possible? Find the GCF of 150 and 120. The common prime factors are 2, 3, and 5.

49. Example 4-3a Answer: So, 30 plates are possible. The GCF of 150 and 120 is or 30.

50. Example 4-3b Cookie Sale Parents donated 150 chocolate chip cookies and 120 molasses cookies for a school bake sale. How many chocolate chip and molasses cookies will be on each plate? Answer: So, each plate will have 5 chocolate chip cookies and 4 molasses cookies.

51. Example 4-3c Apples There are 96 red apples and 72 green apples to be placed in baskets. a. If the apples are arranged in baskets, and each basket has the same number of red apples and the same number of green apples, what is the largest number of baskets possible? b. How many red apples and green apples will be in each basket? Answer: 24 baskets Answer: 4 red apples and 3 green apples

52. Example 4-4a Find the GCF ofand. Completely factor each expression. Answer: The GCF of is Circle the common factors.

53. Example 4-4b Find the GCF ofand. Answer:

54. Example 4-5a Factor. First, find the GCF of 3x and 12. Now, write each term as a product of the GCF and its remaining factors. Answer: Distributive Property The GCF is 3.

55. Example 4-5b Factor. Answer:

56. End of Lesson 4

57. Lesson 5 Contents Example 1Simplify Fractions Example 2Simplify Fractions Example 3Simplify Fractions in Measurement Example 4Simplify Algebraic Fractions Example 5Simplify Algebraic Fractions

58. Example 5-1a Write in simplest form. The GCF of 16 and 24 is or 8. Answer: Factor the numerator. Factor the denominator. Divide the numerator and denominator by the GCF. Simplest form

59. Example 5-1b Write in simplest form. Answer:

60. Example 5-2a Write in simplest form. Simplify. Answer: Divide the numerator and the denominator by the GCF,. 1111 1111

61. Example 5-2b Write in simplest form. Answer:

62. Example 5-3a Measurement 250 pounds is what part of 1 ton? There are 2000 pounds in 1 ton. Write the fraction in simplest form. Answer: So, 250 pounds is of a ton. Simplify. Divide the numerator and the denominator by the GCF,. 1111 1111

63. Example 5-3b 80 feet is what part of 40 yards? Answer:

64. Example 5-4a Simplify. Answer: 11 Divide the numerator and the denominator by the GCF,. 111 1

65. Example 5-4b Simplify. Answer:

66. Example 5-5a ABCDABCD Read the Test Item In simplest form means that the GCF of the numerator and denominator is 1. Which fraction is written in simplest form? Multiple-Choice Test Item

67. Example 5-5b Solve the Test Item Answer: C Factor. 1111 1111

68. Example 5-5c ABCDABCD Answer: D Which fraction is written in simplest form? Multiple-Choice Test Item

69. End of Lesson 5

70. Lesson 6 Contents Example 1Multiply Powers Example 2Multiply Monomials Example 3Divide Powers Example 4Divide Powers to Solve a Problem

71. Example 6-1a Find. Check Answer: Add the exponents. The common base is 3.

72. Example 6-1b Find. Answer:

73. Find. Example 6-2a Answer: The common base is y. Add the exponents.

74. Example 6-2b Find (3p 4 )(–2p 3 ). Answer: –6p 7 Use the Commutative and Associative Properties. (3p 4 )(–2p 3 )(3 –2)(p 4 p 3 ) Add the exponents. –6p 7 The common base is p. (–6)(p 4+3 )

75. Example 6-2c Find each product. Answer: a. b. Answer:

76. Example 6-3a The common base is 8. Answer: Subtract the exponents. Find.

77. Example 6-3b The common base is x. Subtract the exponents. Answer: Find.

78. Example 6-3c Find Answer:

79. Example 6-4a Folding Paper If you fold a sheet of paper in half, you have a thickness of 2 sheets. Folding again, you have a thickness of 4 sheets. Continue folding in half and recording the thickness. How many times thicker is a sheet that has been folded 4 times than a sheet that has not been folded? Write a division expression to compare the thickness. Subtract the exponents. Answer: So, the paper is 16 times thicker.

80. Example 6-4b Racing Car A can run at a speed ofmiles per hour and car B runs at a speed of miles per hour. How many times faster is car A than car B? Answer: Car A is 2 times faster than car B.

81. End of Lesson 6

82. Lesson 7 Contents Example 1Use Positive Exponents Example 2Use Negative Exponents Example 3Use Exponents to Solve a Problem Example 4Algebraic Expressions with Negative Exponents

83. Example 7-1a Answer: Definition of negative exponent Write using a positive exponent.

84. Example 7-1b Definition of negative exponent Answer: Write using a positive exponent.

85. Example 7-1c Write each expression using a positive exponent. Answer: a. b. Answer:

86. Example 7-2a Write as an expression using a negative exponent. Answer: Find the prime factorization of 125. Definition of exponents Definition of negative exponent

87. Example 7-2b Write as an expression using a negative exponent. Answer:

88. Example 7-3a Physics An atom is an incredibly small unit of matter. The smallest atom has a diameter of approximately of a nanometer, or 0.0000000001 meter. Write the decimal as a fraction and as a power of 10. Answer: Write the decimal as a fraction. Definition of negative exponent

89. Example 7-3b Write 0.000001 as a fraction and as a power of 10. Answer:

90. Example 7-4a Find. Answer: Replace r with –4. Definition of negative exponent Evaluate.if

91. Example 7-4b Answer: Evaluate. if

92. End of Lesson 7

93. Lesson 8 Contents Example 1Express Numbers in Standard Form Example 2Express Numbers in Scientific Notation Example 3Use Scientific Notation to Solve a Problem Example 4Compare Numbers in Scientific Notation

94. Express in standard form. Example 8-1a Answer: 43,950 Move the decimal point 4 places to the right.

95. Example 8-1b Answer: 0.00000679 Move the decimal point 6 places to the left. Express in standard form.

96. Example 8-1c Express each number in standard form. a. b. Answer: 2,614,000 Answer: 0.000803

97. Example 8-2a Express 800,000 in scientific notation. Answer: The exponent is positive. The decimal point moves 5 places.

98. Example 8-2b Express 1,320,000 in scientific notation. The exponent is positive. The decimal point moves 6 places. Answer:

99. Example 8-2c Express 0.0119 in scientific notation. The exponent is negative. The decimal point moves 2 places. Answer:

100. Example 8-2d Express each number in scientific notation. a. 65,000 b. 3,024,000 c. 0.00042 Answer:

101. Example 8-3a Space The table shows the planets and their distances from the Sun. Estimate how many times farther Pluto is from the Sun than Mercury is from the Sun. PlanetDistance from the Sun (km) Mercury 5.80 x 10 7 Venus 1.03 x 10 8 Earth1.55 x 10 8 Mars2.28 x 10 8 Jupiter7.78 x 10 8 Saturn1.43 x 10 9 Uranus2.87 x 10 9 Neptune4.50 x 10 9 Pluto5.90 x 10 9

102. Example 8-3a Explore You know that the distance from the Sun to Pluto iskm and the distance from the Sun to Mercury iskm. Plan To find how many times farther Pluto is from the Sun than Mercury is from the Sun, find the ratio of Pluto’s distance to Mercury’s distance. Since you are estimating, round the distance to and round the distance to.

103. Example 8-3b Examine Use estimation to check the reasonableness of the results. Solve Divide Answer: So, Pluto is about 1.0  10 2 or 100 times farther from the Sun than Mercury.

104. PlanetDistance from the Sun (km) Mercury 5.80 x 10 7 Venus 1.03 x 10 8 Earth1.55 x 10 8 Mars2.28 x 10 8 Jupiter7.78 x 10 8 Saturn1.43 x 10 9 Uranus2.87 x 10 9 Neptune4.50 x 10 9 Pluto5.90 x 10 9 Example 8-3c Space Use the table to estimate how many times farther Pluto is from the Sun than Earth is from the Sun. Answer: 30 times farther

105. Example 8-4a Space The diameters of Mercury, Saturn, and Pluto arekilometers,kilometers, and kilometers, respectively. List the planets in order of increasing diameter. First, order the numbers according to their exponents. Then, order the numbers with the same exponent by comparing the factors.

106. Example 8-4b Answer: So, the order is Pluto, Mercury, and Saturn. Step 1 Step 2 Mercury and Pluto Saturn Pluto Mercury Compare the factors:

107. Example 8-4c Order the numbers,,, andin decreasing order. Answer:,,, and.

108. End of Lesson 8

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