Download Section 1.3 Integer Exponents

Working with Powers
• Integer Exponents
• Exponent Rules
• Order of Operations
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• 2 × 2 = 22
• 2 × 2 × 2 = 23
• 2 × 2 × 2 × 2 = 24
• How can we write these in
shorter notation?
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• Multiplication is a shortcut for
repeated addition.
3+3+3+3+3=5×3
• Exponents are a shortcut for
repeated multiplication.
5
3
3×3×3×3×3=
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Write these numbers using
exponential notation:
• 10,000 = 10?
• 27 = 3?
• 32 = 2?
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Integer Exponents
4
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• Base 4
• Exponent 3
• 43 is called a power
• 43 = 4 x 4 x 4
• = 64
5
You try these . . .
•
2
7
•
4
5
• 210
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Operations with Exponents
• Multiply (x3)∙(x4)
= (x ∙ x ∙ x) ∙ (x ∙ x ∙ x ∙ x)
= (x ∙ x ∙ x ∙ x ∙ x ∙ x ∙ x) = x7
A5 ∙ A4 =
x4
x
xxxx
• Divide: 3  x  x  x  1  x
x
m4
m3
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Exponent Rules
#1
a  1  a  a  ....  a
n
n times
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Zero as an exponent
•
0
4
•
0
9
• 170
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Exponent Rules
#2
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a 1
0
10
a0 = 1
• M0
• (pq)0
• (2x2y)0
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Negative Exponents
•
-1
3
•
-4
2
• 7-1
• (1/2) -1
• 5-2
• (3/4) -2
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Exponent Rules
#3
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a
n
1
n
 n  1 a
a
13
a
n
1
 n
a
• M-2
• x-5
• (1/y)-3
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Combining Exponents
x x
 x x  x x x
2
x
3
5
m
n
a a =?
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Exponent Rules
#4
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a a a
m
n
m n
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a a a
m
n
m n
• x5x4
• y-2y7
•
b3b-6
• m6m-6
•
(2x3)(4x-2)
• 2z-3z5w-6w-2
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Combining Exponents
5
x
2
x
xxxxx

xx
3
x
am/an=?
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Exponent Rules
#5
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m
a
mn

a
n
a
19
m
a
mn
a
n
a
6
x
2
x
2
y
7
y
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5m
4
15m
5
10 p
3
5p
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Combining Exponents
x 
2 3
 x x x
 xxxxxx
2
x
2
2
6
(am)n=?
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Exponent Rules
#6
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(a )  a
m n
mn
22
(a )  a
m n
mn
• (w4)5
• (p-3)4
• (5x4)-2
• (-3y-7)3
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Combining Exponents
3
xy xy xy
 xy 
 
  
z z z
 z 
xxx y y y

zzz
3
3
x y

3
z
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Exponent Rules
#7 (distributive rule for exponents)
n
 ab  a b
   n
c
 c 
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n n
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n
a nb n
 ab 

 
n
c
c


x
2
y
 2m 
 5 
n 
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
3 4
3
5m n 
 4 2
7
 5a b
 3
 c
6
4



2
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Exponent Rules
• 1. a n  a  a  ....  a
• 2.
• 3.
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a 1
0
a
n
1
 n
a
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Exponent Rules
• 4.
a a a
m
m
n
a
mn
• 5.

a
an
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m n
• 6.
(a )  a
m n
mn
n
n n
ab
a


• 7.    b
n
c
c
 
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