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# Integrated Math 2 Sections 2-7 and 2-8

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Properties of Exponents and Zero and Negative Exponents

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### Integrated Math 2 Sections 2-7 and 2-8

1. 1. SECTIONS 2-7 AND 2-8 Properties of Exponents and Zero and Negative Exponents
2. 2. ESSENTIAL QUESTIONS How do you choose appropriate units of measure? How do you evaluate variable expressions? How do you write numbers using zero and negative integers as exponents? How do you write numbers in scientiﬁc notation? Where you’ll see this: Biology, ﬁnance, computers, population, physics, astronomy
3. 3. VOCABULARY 1. Exponential Form: 2. Base: 3. Exponent: 4. Standard Form: 5. Scientiﬁc Notation:
4. 4. VOCABULARY 1. Exponential Form: The form you use to represent multiplying a number by itself numerous times 2. Base: 3. Exponent: 4. Standard Form: 5. Scientiﬁc Notation:
5. 5. VOCABULARY 1. Exponential Form: The form you use to represent multiplying a number by itself numerous times 2. Base: The number that is being multiplied over and over 3. Exponent: 4. Standard Form: 5. Scientiﬁc Notation:
6. 6. VOCABULARY 1. Exponential Form: The form you use to represent multiplying a number by itself numerous times 2. Base: The number that is being multiplied over and over 3. Exponent: The number of times we multiply the base 4. Standard Form: 5. Scientiﬁc Notation:
7. 7. VOCABULARY 1. Exponential Form: The form you use to represent multiplying a number by itself numerous times 2. Base: The number that is being multiplied over and over 3. Exponent: The number of times we multiply the base 4. Standard Form: Any number in decimal form 5. Scientiﬁc Notation:
8. 8. VOCABULARY 1. Exponential Form: The form you use to represent multiplying a number by itself numerous times 2. Base: The number that is being multiplied over and over 3. Exponent: The number of times we multiply the base 4. Standard Form: Any number in decimal form 5. Scientiﬁc Notation: A number with two factors, where the ﬁrst factor is a number ≥ 1 and < 10, and the second is a power of 10.
9. 9. ACTIVITY: PAGE 82 Hours 1 2 3 4 5 # of 2 4 8 16 32 organisms Pattern: Sixth hour: Seventh hour: Eighth hour: How long until there are 2000?
10. 10. ACTIVITY: PAGE 82 Hours 1 2 3 4 5 # of 2 4 8 16 32 organisms Pattern: Sixth hour: Seventh hour: Eighth hour: How long until there are 2000?
11. 11. ACTIVITY: PAGE 82 Hours 1 2 3 4 5 # of 2 4 8 16 32 organisms Pattern: Sixth hour: Seventh hour: Eighth hour: How long until there are 2000?
12. 12. ACTIVITY: PAGE 82 Hours 1 2 3 4 5 # of 2 4 8 16 32 organisms Pattern: Sixth hour: Seventh hour: Eighth hour: How long until there are 2000?
13. 13. ACTIVITY: PAGE 82 Hours 1 2 3 4 5 # of 2 4 8 16 32 organisms Pattern: Sixth hour: Seventh hour: Eighth hour: How long until there are 2000?
14. 14. ACTIVITY: PAGE 82 Hours 1 2 3 4 5 # of 2 4 8 16 32 organisms Pattern: Sixth hour: Seventh hour: Eighth hour: How long until there are 2000?
15. 15. ACTIVITY: PAGE 82 Hours 1 2 3 4 5 # of 2 4 8 16 32 organisms Pattern: The number of organisms doubles Sixth hour: Seventh hour: Eighth hour: How long until there are 2000?
16. 16. ACTIVITY: PAGE 82 Hours 1 2 3 4 5 # of 2 4 8 16 32 organisms Pattern: The number of organisms doubles Sixth hour: 64 Seventh hour: Eighth hour: How long until there are 2000?
17. 17. ACTIVITY: PAGE 82 Hours 1 2 3 4 5 # of 2 4 8 16 32 organisms Pattern: The number of organisms doubles Sixth hour: 64 Seventh hour: 128 Eighth hour: How long until there are 2000?
18. 18. ACTIVITY: PAGE 82 Hours 1 2 3 4 5 # of 2 4 8 16 32 organisms Pattern: The number of organisms doubles Sixth hour: 64 Seventh hour: 128 Eighth hour: 256 How long until there are 2000?
19. 19. ACTIVITY: PAGE 82 Hours 1 2 3 4 5 # of 2 4 8 16 32 organisms Pattern: The number of organisms doubles Sixth hour: 64 Seventh hour: 128 Eighth hour: 256 How long until there are 2000? 11th hour
20. 20. EXAMPLE 1 Evaluate for x = .8 and y = 1.2. 2 3 2 a. 3x y b. (4x) y
21. 21. EXAMPLE 1 Evaluate for x = .8 and y = 1.2. 2 3 2 a. 3x y b. (4x) y = 3(.8) (1.2) 2
22. 22. EXAMPLE 1 Evaluate for x = .8 and y = 1.2. 2 3 2 a. 3x y b. (4x) y = 3(.8) (1.2) 2 = 3(.64)(1.2)
23. 23. EXAMPLE 1 Evaluate for x = .8 and y = 1.2. 2 3 2 a. 3x y b. (4x) y = 3(.8) (1.2) 2 = 3(.64)(1.2) = 2.304
24. 24. EXAMPLE 1 Evaluate for x = .8 and y = 1.2. 2 3 2 a. 3x y b. (4x) y = 3(.8) (1.2) 2 = [4(.8)] (1.2) 3 2 = 3(.64)(1.2) = 2.304
25. 25. EXAMPLE 1 Evaluate for x = .8 and y = 1.2. 2 3 2 a. 3x y b. (4x) y = 3(.8) (1.2) 2 = [4(.8)] (1.2) 3 2 = 3(.64)(1.2) = (3.2) (1.2) 3 2 = 2.304
26. 26. EXAMPLE 1 Evaluate for x = .8 and y = 1.2. 2 3 2 a. 3x y b. (4x) y = 3(.8) (1.2) 2 = [4(.8)] (1.2) 3 2 = 3(.64)(1.2) = (3.2) (1.2) 3 2 = 2.304 = (32.768)(1.44)
27. 27. EXAMPLE 1 Evaluate for x = .8 and y = 1.2. 2 3 2 a. 3x y b. (4x) y = 3(.8) (1.2) 2 = [4(.8)] (1.2) 3 2 = 3(.64)(1.2) = (3.2) (1.2) 3 2 = 2.304 = (32.768)(1.44) = 47.18592
28. 28. PRODUCT RULE b ib = b m n m+n
29. 29. PRODUCT RULE b ib = b m n m+n x ix = 3 5
30. 30. PRODUCT RULE b ib = b m n m+n x ix = xixix 3 5
31. 31. PRODUCT RULE b ib = b m n m+n x ix = xixix ixixixixix 3 5
32. 32. PRODUCT RULE b ib = b m n m+n x ix = xixix ixixixixix = x 3 5 8
33. 33. POWER RULE (b ) = b m n mn
34. 34. POWER RULE (b ) = b m n mn 2 3 (y )
35. 35. POWER RULE (b ) = b m n mn (y ) = y iy iy 2 3 2 2 2
36. 36. POWER RULE (b ) = b m n mn (y ) = y iy iy = y 2 3 2 2 2 6
37. 37. POWER OF A PRODUCT RULE (ab) = a b m m m
38. 38. POWER OF A PRODUCT RULE (ab) = a b m m m 4 (gf )
39. 39. POWER OF A PRODUCT RULE (ab) = a b m m m (gf ) = gf igf igf igf 4
40. 40. POWER OF A PRODUCT RULE (ab) = a b m m m (gf ) = gf igf igf igf = g f 4 4 4
41. 41. QUOTIENT RULE m b b ÷b = n =b m n m−n b
42. 42. QUOTIENT RULE m b b ÷b = n =b m n m−n b 5 c 3 c
43. 43. QUOTIENT RULE m b b ÷b = n =b m n m−n b 5 c cicicicic 3 = c cicic
44. 44. QUOTIENT RULE m b b ÷b = n =b m n m−n b 5 c cicicicic 3 = c cicic
45. 45. QUOTIENT RULE m b b ÷b = n =b m n m−n b 5 c cicicicic 3 = c cicic
46. 46. QUOTIENT RULE m b b ÷b = n =b m n m−n b 5 c cicicicic 3 = c cicic
47. 47. QUOTIENT RULE m b b ÷b = n =b m n m−n b 5 c cicicicic 3 = =c 2 c cicic
48. 48. POWER OF A QUOTIENT RULE m ⎛ a⎞ m a ⎜ b ⎟ = bm ⎝ ⎠
49. 49. POWER OF A QUOTIENT RULE m ⎛ a⎞ m a ⎜ b ⎟ = bm ⎝ ⎠ 3 ⎛ d⎞ ⎜ w⎟ ⎝ ⎠
50. 50. POWER OF A QUOTIENT RULE m ⎛ a⎞ m a ⎜ b ⎟ = bm ⎝ ⎠ 3 ⎛ d⎞ d d d ⎜ w⎟ = wiwiw ⎝ ⎠
51. 51. POWER OF A QUOTIENT RULE m ⎛ a⎞ a m ⎜ b ⎟ = bm ⎝ ⎠ 3 ⎛ d⎞ d d d d 3 ⎜ w ⎟ = w i w i w = w3 ⎝ ⎠
52. 52. EXAMPLE 2 Simplify. 2 5 4 2 a. 3 i3 b. (6m )
53. 53. EXAMPLE 2 Simplify. 2 5 4 2 a. 3 i3 b. (6m ) =3 2+ 5
54. 54. EXAMPLE 2 Simplify. 2 5 4 2 a. 3 i3 b. (6m ) =3 2+ 5 =3 7
55. 55. EXAMPLE 2 Simplify. 2 5 4 2 a. 3 i3 b. (6m ) =3 2+ 5 =3 7 = 2187
56. 56. EXAMPLE 2 Simplify. 2 5 4 2 a. 3 i3 b. (6m ) =3 2+ 5 =6 m 2 4( 2) =3 7 = 2187
57. 57. EXAMPLE 2 Simplify. 2 5 4 2 a. 3 i3 b. (6m ) =3 2+ 5 =6 m 2 4( 2) =3 7 = 36m 8 = 2187
58. 58. EXAMPLE 3 Evaluate for x = 1/2 and y = 2/3. 2 3 2 a. x y b. 3x y
59. 59. EXAMPLE 3 Evaluate for x = 1/2 and y = 2/3. 2 3 2 a. x y b. 3x y =( 2)() 1 2 2 3
60. 60. EXAMPLE 3 Evaluate for x = 1/2 and y = 2/3. 2 3 2 a. x y b. 3x y =( )() 1 2 2 2 3 = ( )( ) 1 4 2 3
61. 61. EXAMPLE 3 Evaluate for x = 1/2 and y = 2/3. 2 3 2 a. x y b. 3x y =( )() 1 2 2 2 3 = ( )( ) 1 4 2 3 = 2 12
62. 62. EXAMPLE 3 Evaluate for x = 1/2 and y = 2/3. 2 3 2 a. x y b. 3x y =( )() 1 2 2 2 3 = ( )( ) 1 4 2 3 = 2 12 = 1 6
63. 63. EXAMPLE 3 Evaluate for x = 1/2 and y = 2/3. 2 3 2 a. x y b. 3x y =( )() = 3( )() 1 2 2 3 2 1 2 2 3 2 3 = ( )( ) 1 4 2 3 = 2 12 = 1 6
64. 64. EXAMPLE 3 Evaluate for x = 1/2 and y = 2/3. 2 3 2 a. x y b. 3x y =( )() = 3( )() 1 2 2 3 2 1 2 2 3 2 3 = ( )( ) 1 4 2 3 = 3 ( )( ) 1 8 4 9 = 2 12 = 1 6
65. 65. EXAMPLE 3 Evaluate for x = 1/2 and y = 2/3. 2 3 2 a. x y b. 3x y =( )() = 3( )() 1 2 2 3 2 1 2 2 3 2 3 = ( )( ) 1 4 2 3 = 3 ( )( ) 1 8 4 9 = 2 12 = 3( 4 72 ) = 1 6
66. 66. EXAMPLE 3 Evaluate for x = 1/2 and y = 2/3. 2 3 2 a. x y b. 3x y =( )() = 3( )() 1 2 2 3 2 1 2 2 3 2 3 = ( )( ) 1 4 2 3 = 3 ( )( )1 8 4 9 = 2 12 = 3( 4 72 ) = 1 6 = 12 72
67. 67. EXAMPLE 3 Evaluate for x = 1/2 and y = 2/3. 2 3 2 a. x y b. 3x y =( )() = 3( )() 1 2 2 3 2 1 2 2 3 2 3 = ( )( ) 1 4 2 3 = 3 ( )( )1 8 4 9 = 2 12 = 3( 4 72 ) = 1 6 = 12 72 = 1 6
68. 68. ZERO PROPERTY OF EXPONENTS b =1 0
69. 69. ZERO PROPERTY OF EXPONENTS b =1 0 4 x 4 x
70. 70. ZERO PROPERTY OF EXPONENTS b =1 0 4 x 4− 4 4 = x x
71. 71. ZERO PROPERTY OF EXPONENTS b =1 0 4 x 4− 4 4 = x =x 0 x
72. 72. ZERO PROPERTY OF EXPONENTS b =1 0 4 x 4− 4 4 = x =x 0 x 4 x 4 x
73. 73. ZERO PROPERTY OF EXPONENTS b =1 0 4 x 4− 4 4 = x =x 0 x 4 x 4 =1 x
74. 74. PROPERTY OF NEGATIVE EXPONENTS 1 b −m = m b
75. 75. PROPERTY OF NEGATIVE EXPONENTS 1 b −m = m b 3 h 7 h
76. 76. PROPERTY OF NEGATIVE EXPONENTS 1 b −m = m b 3 h 7 = h 3−7 h
77. 77. PROPERTY OF NEGATIVE EXPONENTS 1 b −m = m b 3 h −4 7 = h =h 3−7 h
78. 78. PROPERTY OF NEGATIVE EXPONENTS 1 b −m = m b 3 h −4 7 = h =h 3−7 h 3 h 7 h
79. 79. PROPERTY OF NEGATIVE EXPONENTS 1 b −m = m b 3 h −4 7 = h =h 3−7 h 3 h hihih 7 = h hihihihihihih
80. 80. PROPERTY OF NEGATIVE EXPONENTS 1 b −m = m b 3 h −4 7 = h =h 3−7 h 3 h hihih 1 7 = = 4 h hihihihihihih h
81. 81. EXAMPLE 4 Simplify each expression. Write the answer with a positive exponent. 2 x 1 a. 8 3 b. y i 4 −3 2 x c. (z ) y
82. 82. EXAMPLE 4 Simplify each expression. Write the answer with a positive exponent. 2 x 1 a. 8 3 b. y i 4 −3 2 x c. (z ) y =x 2− 8
83. 83. EXAMPLE 4 Simplify each expression. Write the answer with a positive exponent. 2 x 1 a. 8 3 b. y i 4 −3 2 x c. (z ) y =x 2− 8 −6 =x
84. 84. EXAMPLE 4 Simplify each expression. Write the answer with a positive exponent. 2 x 1 a. 8 3 b. y i 4 −3 2 x c. (z ) y =x 2− 8 −6 =x 1 = 6 x
85. 85. EXAMPLE 4 Simplify each expression. Write the answer with a positive exponent. 2 x 1 a. 8 3 b. y i 4 −3 2 x c. (z ) y 3 y =x 2− 8 = 4 y −6 =x 1 = 6 x
86. 86. EXAMPLE 4 Simplify each expression. Write the answer with a positive exponent. 2 x 1 a. 8 3 b. y i 4 −3 2 x c. (z ) y 3 y =x 2− 8 = 4 y −6 =x =y −1 1 = 6 x
87. 87. EXAMPLE 4 Simplify each expression. Write the answer with a positive exponent. 2 x 1 a. 8 3 b. y i 4 −3 2 x c. (z ) y 3 y =x 2− 8 = 4 y −6 =x =y −1 1 1 = 6 = x y
88. 88. EXAMPLE 4 Simplify each expression. Write the answer with a positive exponent. 2 x 1 a. 8 3 b. y i 4 −3 2 x c. (z ) y 3 −6 =x 2− 8 = 4 y =z y −6 =x =y −1 1 1 = 6 = x y
89. 89. EXAMPLE 4 Simplify each expression. Write the answer with a positive exponent. 2 x 1 a. 8 3 b. y i 4 −3 2 x c. (z ) y 3 −6 =x 2− 8 = 4 y =z y =x −6 1 =y −1 = 6 1 1 z = 6 = x y
90. 90. EXAMPLE 5 Evaluate each expression when m = -2 and n = 4. 4 3 −2 5 −3 a. 6m b. (n ) c. m n
91. 91. EXAMPLE 5 Evaluate each expression when m = -2 and n = 4. 4 3 −2 5 −3 a. 6m b. (n ) c. m n = 6(−2) 4
92. 92. EXAMPLE 5 Evaluate each expression when m = -2 and n = 4. 4 3 −2 5 −3 a. 6m b. (n ) c. m n = 6(−2) 4 = 6(16)
93. 93. EXAMPLE 5 Evaluate each expression when m = -2 and n = 4. 4 3 −2 5 −3 a. 6m b. (n ) c. m n = 6(−2) 4 = 6(16) = 96
94. 94. EXAMPLE 5 Evaluate each expression when m = -2 and n = 4. 4 3 −2 5 −3 a. 6m b. (n ) c. m n −6 = 6(−2) 4 =n = 6(16) = 96
95. 95. EXAMPLE 5 Evaluate each expression when m = -2 and n = 4. 4 3 −2 5 −3 a. 6m b. (n ) c. m n −6 = 6(−2) 4 =n = 6(16) 1 = 6 n = 96
96. 96. EXAMPLE 5 Evaluate each expression when m = -2 and n = 4. 4 3 −2 5 −3 a. 6m b. (n ) c. m n −6 = 6(−2) 4 =n = 6(16) 1 = 6 n = 96 1 = 6 4
97. 97. EXAMPLE 5 Evaluate each expression when m = -2 and n = 4. 4 3 −2 5 −3 a. 6m b. (n ) c. m n −6 = 6(−2) 4 =n = 6(16) 1 = 6 n = 96 1 1 = 6 = 4 4096
98. 98. EXAMPLE 5 Evaluate each expression when m = -2 and n = 4. 4 3 −2 5 −3 a. 6m b. (n ) c. m n −6 −3 = 6(−2) 4 =n = (−2) (4) 5 = 6(16) 1 = 6 n = 96 1 1 = 6 = 4 4096
99. 99. EXAMPLE 5 Evaluate each expression when m = -2 and n = 4. 4 3 −2 5 −3 a. 6m b. (n ) c. m n −6 −3 = 6(−2) 4 =n = (−2) (4) 5 = 6(16) 1 (−2) 5 = 6 = 3 n (4) = 96 1 1 = 6 = 4 4096
100. 100. EXAMPLE 5 Evaluate each expression when m = -2 and n = 4. 4 3 −2 5 −3 a. 6m b. (n ) c. m n −6 −3 = 6(−2) 4 =n = (−2) (4) 5 = 6(16) 1 (−2) 5 = 6 = 3 n (4) = 96 1 1 −32 = 6 = = 4 4096 64
101. 101. EXAMPLE 5 Evaluate each expression when m = -2 and n = 4. 4 3 −2 5 −3 a. 6m b. (n ) c. m n −6 −3 = 6(−2) 4 =n = (−2) (4) 5 = 6(16) 1 (−2) 5 = 6 = 3 n (4) = 96 1 1 −32 −1 = 6 = = = 4 4096 64 2
102. 102. EXAMPLE 6 Write in scientiﬁc notation. a. .0000013 b. 230,000,000,000
103. 103. EXAMPLE 6 Write in scientiﬁc notation. a. .0000013 b. 230,000,000,000 -6 =1.3i10
104. 104. EXAMPLE 6 Write in scientiﬁc notation. a. .0000013 b. 230,000,000,000 -6 11 =1.3i10 =2.3i10
105. 105. EXAMPLE 7 Write in standard form. 6 −9 a. 7.2i10 b. 3.5i10
106. 106. EXAMPLE 7 Write in standard form. 6 −9 a. 7.2i10 b. 3.5i10 = 7, 200,000
107. 107. EXAMPLE 7 Write in standard form. 6 −9 a. 7.2i10 b. 3.5i10 = 7, 200,000 = .0000000035
108. 108. EXAMPLE 8 −24 The mass of one hydrogen atom is 1.67i10 grams. Find the mass of 2,700,000,000,000,000 hydrogen atoms.
109. 109. EXAMPLE 8 −24 The mass of one hydrogen atom is 1.67i10 grams. Find the mass of 2,700,000,000,000,000 hydrogen atoms. −24 15 (1.67i10 )(2.7i10 )
110. 110. EXAMPLE 8 −24 The mass of one hydrogen atom is 1.67i10 grams. Find the mass of 2,700,000,000,000,000 hydrogen atoms. −24 15 (1.67i10 )(2.7i10 ) −24 = (1.67i2.7)(10 i10 15 )
111. 111. EXAMPLE 8 −24 The mass of one hydrogen atom is 1.67i10 grams. Find the mass of 2,700,000,000,000,000 hydrogen atoms. −24 15 (1.67i10 )(2.7i10 ) −24 = (1.67i2.7)(10 i10 15 ) −9 = 4.509i10
112. 112. EXAMPLE 8 −24 The mass of one hydrogen atom is 1.67i10 grams. Find the mass of 2,700,000,000,000,000 hydrogen atoms. −24 15 (1.67i10 )(2.7i10 ) −24 = (1.67i2.7)(10 i10 15 ) −9 = 4.509i10 = .000000004509 gram
113. 113. HOMEWORK
114. 114. HOMEWORK p. 84 #1-48 multiples of 3; p. 88 #1-48 multiples of 3 “When you’re screwing up and nobody’s saying anything to you anymore, that means they gave up [on you]...When you see yourself doing something badly and nobody’s bothering to tell you anymore, that’s a very bad place to be. Your critics are the ones who are telling you they still love and care.” - Randy Pausch