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#1
An exponent tells how many times a number is
multiplied by itself.
Base
Exponent
3
8
Factored Form
8•8•8 = 512
Exponential Form
3•3•a•a•a
2 3
3 a
x•y•x•2
2 1
2x y

2 2 4
9
3 3
2
3
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2
#2
When simplifying exponents you must
watch the sign and the parenthesis!
2
5
=
2
(-5)
5•5
=
25
=(-5)(-5) =
2
–5
=
2
–5•5
=
–25
25 –(5) = -1(5)(5) = –25
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Evaluate The Power
#3
1)
2)
5
2
2
4
 55
 25
3)
 2  2  2  2 4)
 16
10
3
12
2
4
 10  10  10
 1000
 12  12
 144
To find 5 on my calculator I type in
5
^ 4 = 625
5 yx 4 = 625
3
Try to find 9 = 729
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Powers of Ten
#4
1
10
10
2
3
10
-1
10
100
-2
1,000 10
4
10 10,000
-3
5
10
10 100,000
10
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1
1
10
= 0.1
1
10 2
1
100
1
10 3
1
=
1000
= 0.01
0.001
Negative Exponents
-n
#5
For any integer n, a is the
reciprocal of an
EXAMPLES:
a
n
1
 n
a
3
A negative exponent
is an inverse!
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2
1
 2
3
1
(5) 
4
(5)
4
#6
Any number to the zero
power is ALWAYS ONE.
x0 = 1
Ex:
4 1
0
5
2
5
2
 3  4  1
5
2 2
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0

1
5 1
0
#7 Exponents and Parenthesis
Factored Form
8•x•x•x
(8x)(8x)(8x)
4(xy)(xy)
(5 x x x)(5 x x x)
(2 y y z) (2 y y z)
8x
Exponential Form
3
(8x)
3
4(xy)
=
2
3 2
(5x )
=
=
2 2
3 3
8 x
512x
2 2
4x y
5
2
3 2=
(x )
2 4 2
(2y z) = 2 y z
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=
3
25x
= 4y
6
4 2
z
# 8 Fractions With Exponents
1
1
1

1) 3 
8
222
2

2
4
2
2
2


2)
9
3 3
3
 
 
2
1
3
4)

3
1
2
 
2
9
2
25
2
5

5)

4
5
2
2
1

1

1
1
1
1
6)
9



3) 
2 
25
5 5
5
9
81
2
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#9
Negative Exponent Examples
1
1) n  5
n
5
3 4
2) a b
1
= 3 4
ab
1
3) m n  3  1
m
1
 3
m
3 0
1
3
4) 3a  3  4  4
a
a
4
1
1
5)  3a  
4 
4
 3a  81a
4
1
6)  5x  5  2
x

5
 2
x
2
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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
3
3 2•3 3
(5 a b) = 5 a
1 3 5
(2 x )
1 8 2
6 3
b = 125a b
5 3•5
=2 x
15
= 32x
2 8•2 2
8(3 y z) = 8 (3 y
16 2
z ) = 72y z
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#13
Product of a Power Property
8
x •x
5
=x
8+5
=x
13
Power of a Power Property
7
(4a b)
3
3 7•3 3
=4 a
b = 64a21b3
Power of a Product Property
2
2
(-3 • 4) = (-12) = (-12)(-12) =144
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#14
Prime Factorization is when you write a
number as the product of prime numbers.
36
Circle the
prime
numbers
2
 Factor Tree
18
2 9
3
3
36  2  2  3  3
2
2
36  2  3
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#15
Factoring
12
2
2 23
6
2
3
2m
10m
2

5

m

m

m

m
1)
=
=
3 5 m
3
15m
4
3
3
2
3a
27a b
3•3•3•
a
•
a
•
a
•
b
2)
=
2 =
2• 2•3•3• a • b • b
4b
36ab
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# 16
When Dividing Powers with the same base,
just SUBTRACT THE EXPONENTS. This is called the
Quotient of Powers Property.
x = x  x  x  x  x = x  x  x  x3
2
xx
x
5
or
5
x = x 5 2  x 3
2
x
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#17
2
15x y
12xy
3
3
5
12 3 2
15x y
3 12xy
4
15 21 31
y
x
12
5 y2
x
4
5xy
4
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2
# 18
COEFFICIENT:
The number in front of the variable is the
coefficient. Multiply coefficients. Add
exponents if the bases are the same
2x  4x
3
5
2
1) (7ab)(2a )
8 x
3 5
 8x
8
2) (2x y )(3x y)
2
3
3
4
6 x y
14 a b
6 4
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# 19
Dividing Powers With Negatives
Quotient of Powers Property
a
x = x a b
b
x
3
3
3
3
6x
3x
4 =
4
8x
4x
3x  x
=
4
3
4
3x
=
4
3 7
3 3  4
6x
3x
x
x
=
=
=
4
4
4
4
8x
4x
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7
#20
2x  5x
3
2  5  x34  10x 7
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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3
# 21
Quotient of Powers Property
Quotient of Powers Property
a
x = x a b
b
x
2
7  723  75  1
5
3
7
7
2
7 7
3
Same
1  73  1  1
1
2
2
3  5
7
7 7
7
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Extras
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Exponents & Powers
#1
An exponent or power tells how many
times a number is multiplied by itself.
Base
5
2
7
3
4
3
Exponent
“Five to the 2nd power”
“Five squared”
“Seven to the 3rd power”
“Seven cubed”
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55
777
#
Multiplying Powers:
If bases are the same add exponents.
x x
7
x
4
7 4
x
11
Power of a Power:
Used when exponents are on the outside of
parenthesis, just multiply exponents.
2 4 3
(2a b )
2 3
2 a b
3
4 3
 8a b
6 12
Coefficients:
The number in front of the variable is the
coefficient. Multiply coefficients.
2x  4x
3
5
 8x
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8
#
Dividing Powers:
If bases are the same subtract exponents.
1512
6
15
6
12
6
6
3
Negative Exponent:
To get rid of a negative exponent flip it over!
1  1
2  24 16
4
Zero Exponent:
Anything to the zero power is always one!
329 
0
1
copyright©amberpasillas2010