Download Multiplying Powers

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Exponents give us many shortcuts for
multiplying and dividing quickly.
Each of the key rules for exponents has an
importance in algebra.
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5
2 =2•2•2•2•2=?
We know
2•2=4
4•2=8
8 • 2 = 16
16 • 2 = 32
So 2 • 2 • 2 • 2 • 2 = 32
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An exponent tells how many
times a number is multiplied by
itself.
3
Base
4
Exponent
4•4•4 = 64
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How do we write in exponential form?
3 3 3 4  4  4  4
Answer: 3  4
3
4
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How do we write in exponential form?
2  2  2  3 3 4
Answer: 2  3  4
3
2
Notice: 4  4
1
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1
Write each in Exponential Form.
x x x x y y x y
4
2 2 2 2
  
3 3 3 3

3x3x x y
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2
 
3
2
4
3 x y
2
3
Write each in Factored Form.
8a b
8 a a a b b
 xy 
 xy xy xy xy
3 2
4
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If x3 means x • x • x
and x4 means x • x • x • x
3
4
then what is x • x ?
x•x•x•x•x•x•x
7
=x
Can you think of a quick way to come
up with the solution?
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Just Add the Exponents!
3
x x
4
x
3 4
x
7
Your shortcut is called the
Product of Powers Property.
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When multiplying powers with the
same base, just ADD the exponents.
For all positive integers m and n:
m
n
m n
a a  a
Ex:
4
3
2
4 =4
3+2
4 4 4 4 4=4
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5
=
4
5
Try This One!
What is 31 • 34 • 35?
Since we are multiplying like bases
just add the exponents.
(1
+
4
+
5)
Answer: 3
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= 310
Simplify.
2 3
1) 2  2  2
5
2
 32
2
3
7 4
2) d  d  d
11
d
7
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4
Simplify.
21
3) 3  3  3
3
3
 27
2
1
  
1
1
1
1
1
      
2 2 2 2 2
2
3
1
1
1
4)


2
2
2
1

32
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5
Simplify.
5)  2    2    2 
6
  2 
3
33
3
 64
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6) a  b  a
57
2
 a b
12 2
a b
5
2
7
5 2
2
5 2
= 7 (b )
10
= 49b
3 2
2
3 2
(7b )
5
5
(7b )(7b )
(5x )
3
3
(5x )(5x )
= 5 (x )
6
= 25x
Can you think of a quick way to
come up with the solution?
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Just Multiply the Exponents!
(x )  x
3 4
3 4
x
12
Your short cut is called the
Power of a Power Property.
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Just Multiply the Exponents!
2
3
(a )
=
2
a •
2
a •
2
a
=
2+2+2
a
=
Your short cut is called the
Power of Power Property.
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6
a
To find the power of a power, you MULTIPLY the
exponents. This is used when an exponent
is on the outside of parenthesis.
1 2
(5 a b)
3
3 2•3 3
5 a
b
6 3
125a b
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To find the power of a power, you MULTIPLY the
exponents. This is used when an exponent
is on the outside of parenthesis.
1 3 5
(2 x )
5 3•5
2 x
32x
15
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1 5 2
6(3 y z)
2 5•2 2
6(3 y
6(9y
z )
5•2 2
z )
10 2
54y z
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Simplify Using What You Just Learned
2
5
4 5
2) 3m  1m
1) (y )
y
20
3m
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7
Simplify Using What You Just Learned
3) a  a
4
a
3
4) (4x 2 y)2
7
4 2
16 x y
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Simplify Using What You Just Learned
5)
m m
5
6
6)
x
4
y

2 5
20 10
11
m
x y
1
1
20 10
x y
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Simplify Using What You Just Learned
2 4 3
7) (2a b )
8) 2x  4x
3
6 12
8a b
8x
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8
5
Take Out Your Study Guide!!!
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#10
Just flip the fraction over to make the
exponent positive!
1
 
8
2
4
 
7
2
3
2
2
8  8
 
2
1
1
2
 64
2
7  7
 
2
4
4
49

16
 1 
64
 4  4
     (1)3 
 4 
1
 1 
3
3
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 64
#11
When multiplying powers with the
same base, just ADD the exponents.
For all positive integers m and n:
m
a •a
Ex :
2
n
m+n
= a
3
(3 )(3 ) = (3 • 3) • (3 • 3 •3)
=3
5
2+3
4
=3
5
(x )(x ) = x
5+4
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=x
9
# 12
To find the power of a power, you MULTIPLY the
exponents . This is used when an exponent
is on the outside of parenthesis.
1 2
(5 a b) = 5 a
1 3 5
(2 x )
1 8
8(3 y z)
6 3
3 2•3 3
3
b = 125a b
5 3•5
=2 x
2
15
= 32x
2 8•2 2 =
= 8 (3 y z )
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16 2
72y z
Extra slides
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#11
2x  5x
3
7
34

2  5 x  10x
Simplify.
1) (8a )  (3a )
5
7
24 a
2
1 5
3
3) (9x y )(-2xy )
18 x y
3 8
12
2) (-3a)  (4a )
7
1
12 a
4
2 1 3
1 5
4) (6a bc )(5ab )
3
8
6
30 a b c
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