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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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1
25 = 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.
Base
4
3
Exponent
4•4•4 = 64
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2
How do we write in exponential form?
3•3•3• 4 • 4 • 4 • 4
Answer: 33 • 4 4
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How do we write in exponential form?
2 • 2 • 2 •3•3• 4
Answer: 23 • 32 • 41
Notice: 41 = 4
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3
Write each in Exponential Form.
4 2
xi xi xi xi y i y = x y
2 2 2 2
• • •
3 3 3 3
=
3i xi3i xi xi y
2
 
3
4
= 32 x 3 y
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Write each in Factored Form.
8a 3b 2
= 8ia iaia ibib
( xy )
= xy i xy i xy i xy
4
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4
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 ix
4
=x
3+ 4
=x
7
Your shortcut is called the
Product of Powers Property.
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5
When multiplying powers with the
same base, just ADD the exponents.
For all positive integers m and n:
a m • a n = a m+n
Ex:
4
3
2
4 =4
3+2
4 4 4 4 4=4
=
4
5
5
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Try This One!
What is 31 • 34 • 35?
Since we are multiplying like bases
just add the exponents.
Answer:
3(1 + 4 + 5) = 310
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6
Simplify.
1) 22 • 23 = 22+3
= 25
= 32
2) d 7 • d 4 = d 7+ 4
= d11
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Simplify.
2+1
3) 32 • 3 = 3
= 33
= 27
1
() () ()
( )( )
2
3
4) 1 • 1 = 1
2
2
2
= 1•1 • 1•1•1
2 2 2 2 2
= 1
32
5
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7
Simplify.
5) ( −2) • ( −2) = ( −2)
6
= ( −2)
3
3+3
3
6) a 5 • b 2 • a 7
= a 5+ 7 • b 2
= a12 b 2
= 64
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5 2
= 7 (b )
10
= 49b
3 2
= 5 (x )
6
= 25x
(7b )
5
5
(7b )(7b )
(5x )
3
3
(5x )(5x )
2
5 2
2
3 2
Can you think of a quick way to
come up with the solution?
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8
Just Multiply the Exponents!
3 4
12
3• 4
(x ) = x
=x
Your short cut is called the
Power of a Power Property.
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Just Multiply the Exponents!
(a2)3 = a2 • a2 • a2 = a2+2+2
=
a6
Your short cut is called the
Power of Power Property.
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9
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
15
32x
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10
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
4 5
2) 3m 2 • 1m 5
1) (y )
y 20
3 m7
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11
Simplify Using What You Just Learned
3)
a 4 • a3
4) (−4x 2 y) 2
a7
16 x 4 y 2
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Simplify Using What You Just Learned
5)
m •m
5
6
6)
(x
−4
y
)
−2 5
x −20 y −10
m11
1
1
x 20 y10
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12
Simplify Using What You Just Learned
7) (2a 2 b 4 )3
8) 2x 3 • 4x 5
8 a 6 b12
8x 8
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Take Out Your Study Guide!!!
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13
#10
Just flip the fraction over to make the
exponent positive!
1
 
8
4
 
7
−2
2
2
8 = 8
= 
12
1
−2
2
2
7 = 7
= 
42
4
= 64
=
49
16
−3
3
3
 −1 
64
 4 = 4
=
=
 
  (−1)3
 4 
−1
 −1 
= −64
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#11
When multiplying powers with the
same base, just ADD the exponents.
For all positive integers m and n:
m
n
a •a
Ex :
2
m+n
= a
3
(3 )(3 ) = (3 • 3) • (3 • 3 •3)
2+3
=3
5
4
=3
5
5+4
(x )(x ) = x
9
=x
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14
# 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
3
(5 a b) = 5 a
1 3 5
5 3•5
(2 x )
1 8
=2 x
2
8(3 y z)
6 3
3 2•3 3
b = 125a b
15
= 32x
2 8•2 2 =
= 8 (3 y z )
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16 2
72y z
Extra slides
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15
#11
2x 3 • 5x 4
Simplify.
2 • 5 • x 3+ 4 = 10x 7
1) (8a 5 ) • (3a 7 )
1
3) (9x 2 y3 )(-2xy5 )
−18 x 3 y8
24 a12
2) (-3a)1 • (4a 7 )
1
1
4) (6a 2 bc3 )(5ab5 )
−12 a 8
30 a 3 b6 c3
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16