Properties of Exponents and Zero and Negative Exponents

1. SECTIONS 2-7 AND 2-8
Properties of Exponents and Zero and Negative
Exponents

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. VOCABULARY
1. Exponential Form:
2. Base:
3. Exponent:
4. Standard Form:
5. Scientific Notation:

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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. EXAMPLE 1
Evaluate for x = .8 and y = 1.2.
2 3 2
a. 3x y b. (4x) y

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. 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. 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. 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. 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. 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. 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. PRODUCT RULE
b ib = b
m n m+n

29. PRODUCT RULE
b ib = b
m n m+n
x ix =
3 5

30. PRODUCT RULE
b ib = b
m n m+n
x ix = xixix
3 5

31. PRODUCT RULE
b ib = b
m n m+n
x ix = xixix ixixixixix
3 5

32. PRODUCT RULE
b ib = b
m n m+n
x ix = xixix ixixixixix = x
3 5 8

33. POWER RULE
(b ) = b
m n mn

34. POWER RULE
(b ) = b
m n mn
2 3
(y )

35. POWER RULE
(b ) = b
m n mn
(y ) = y iy iy
2 3 2 2 2

36. POWER RULE
(b ) = b
m n mn
(y ) = y iy iy = y
2 3 2 2 2 6

37. POWER OF A PRODUCT
RULE
(ab) = a b
m m m

38. POWER OF A PRODUCT
RULE
(ab) = a b
m m m
4
(gf )

39. POWER OF A PRODUCT
RULE
(ab) = a b
m m m
(gf ) = gf igf igf igf
4

40. POWER OF A PRODUCT
RULE
(ab) = a b
m m m
(gf ) = gf igf igf igf = g f
4 4 4

41. QUOTIENT RULE
m
b
b ÷b = n =b
m n m−n
b

42. QUOTIENT RULE
m
b
b ÷b = n =b
m n m−n
b
5
c
3
c

43. QUOTIENT RULE
m
b
b ÷b = n =b
m n m−n
b
5
c cicicicic
3
=
c cicic

44. QUOTIENT RULE
m
b
b ÷b = n =b
m n m−n
b
5
c cicicicic
3
=
c cicic

45. QUOTIENT RULE
m
b
b ÷b = n =b
m n m−n
b
5
c cicicicic
3
=
c cicic

46. QUOTIENT RULE
m
b
b ÷b = n =b
m n m−n
b
5
c cicicicic
3
=
c cicic

47. QUOTIENT RULE
m
b
b ÷b = n =b
m n m−n
b
5
c cicicicic
3
= =c 2
c cicic

48. POWER OF A QUOTIENT
RULE
m
⎛ a⎞ m
a
⎜ b ⎟ = bm
⎝ ⎠

49. POWER OF A QUOTIENT
RULE
m
⎛ a⎞ m
a
⎜ b ⎟ = bm
⎝ ⎠
3
⎛ d⎞
⎜ w⎟
⎝ ⎠

50. POWER OF A QUOTIENT
RULE
m
⎛ a⎞ m
a
⎜ b ⎟ = bm
⎝ ⎠
3
⎛ d⎞ d d d
⎜ w⎟ = wiwiw
⎝ ⎠

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. EXAMPLE 2
Simplify.
2 5 4 2
a. 3 i3 b. (6m )

53. EXAMPLE 2
Simplify.
2 5 4 2
a. 3 i3 b. (6m )
=3 2+ 5

54. EXAMPLE 2
Simplify.
2 5 4 2
a. 3 i3 b. (6m )
=3 2+ 5
=3 7

55. EXAMPLE 2
Simplify.
2 5 4 2
a. 3 i3 b. (6m )
=3 2+ 5
=3 7
= 2187

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. 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. EXAMPLE 3
Evaluate for x = 1/2 and y = 2/3.
2 3 2
a. x y b. 3x y

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. 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. 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. 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. 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. 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. 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. 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. 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. ZERO PROPERTY OF
EXPONENTS
b =1
0

69. ZERO PROPERTY OF
EXPONENTS
b =1
0
4
x
4
x

70. ZERO PROPERTY OF
EXPONENTS
b =1
0
4
x 4− 4
4 = x
x

71. ZERO PROPERTY OF
EXPONENTS
b =1
0
4
x 4− 4
4 = x =x 0
x

72. ZERO PROPERTY OF
EXPONENTS
b =1
0
4
x 4− 4
4 = x =x 0
x
4
x
4
x

73. ZERO PROPERTY OF
EXPONENTS
b =1
0
4
x 4− 4
4 = x =x 0
x
4
x
4 =1
x

74. PROPERTY OF NEGATIVE
EXPONENTS
1
b −m
= m
b

75. PROPERTY OF NEGATIVE
EXPONENTS
1
b −m
= m
b
3
h
7
h

76. PROPERTY OF NEGATIVE
EXPONENTS
1
b −m
= m
b
3
h
7 = h
3−7
h

77. PROPERTY OF NEGATIVE
EXPONENTS
1
b −m
= m
b
3
h −4
7 = h =h
3−7
h

78. PROPERTY OF NEGATIVE
EXPONENTS
1
b −m
= m
b
3
h −4
7 = h =h
3−7
h
3
h
7
h

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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. EXAMPLE 5
Evaluate each expression when m = -2 and n = 4.
4 3 −2 5 −3
a. 6m b. (n ) c. m n

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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. EXAMPLE 6
Write in scientific notation.
a. .0000013 b. 230,000,000,000

103. EXAMPLE 6
Write in scientific notation.
a. .0000013 b. 230,000,000,000
-6
=1.3i10

104. EXAMPLE 6
Write in scientific notation.
a. .0000013 b. 230,000,000,000
-6 11
=1.3i10 =2.3i10

105. EXAMPLE 7
Write in standard form.
6 −9
a. 7.2i10 b. 3.5i10

106. EXAMPLE 7
Write in standard form.
6 −9
a. 7.2i10 b. 3.5i10
= 7, 200,000

107. EXAMPLE 7
Write in standard form.
6 −9
a. 7.2i10 b. 3.5i10
= 7, 200,000 = .0000000035

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. 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. 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. 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. 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. HOMEWORK

114. HOMEWORK
p. 84 #1-48 multiples of 3; p. 88 #1-48 multiples of 3
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