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Basic Definitions
Exponent - tells how many times the base is used as a
factor.
Base – the number or variable that undergoes repeated
multiplication.
2
BASE
4
EXPONENT
means
2  2  2  2  16
IMPORTANT EXAMPLES
 3 means  (3  3  3  3)  81
4
(3) 4 means (3)  (3)  (3)  (3)  81
 3 means  (3  3  3)  27
3
(3) means (3)  (3)  (3)  27
3
Variable Expressions
x 5 means (use parentheses for multiplica tion ) ( x)( x)( x)( x)( x)
y 3 means ( y )( y )( y )
Here are a few reminders when
evaluating exponents
• A base with an exponent of 0 equals 1
• 100 = 1
• 25,000,0000 = 1
• A base with an exponent of 1 equals the base number
• 51 = 5
• 1,000,0001 = 1,000,000
• A positive base with a positive exponent equals a positive number.
• 52 = 25
• (1/2)2 = ¼
• A negative base with an even exponent equals a positive number.
• (-3)2 = (-3)(-3) = 9
• A negative base with an odd exponent equals a negative number.
• (-3)3 = (-3)(-3)(-3) = -27
• A base with a negative sign in front equals a negative number.
• -33 = -(3 *3*3) = -27
• -92 = -(9*9) = -81
Examples:
•
2 2
( )
3
2
3
2
3
– Write out 𝑡ℎ𝑒 𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛 twice... ( ) * ( )
– Multiply your numerators
– Multiply your denominators
– Final Answer =
• Example:
4
9
1 4
( )
2
– Write the fraction 4 times
– Multiply the numerators
– Multiply the denominators
– Final Answer =
1
16
Multiplying Numbers with Exponents
• To multiply numbers with exponents that have
the same base, add the exponents and keep
the base the same.
• Example: 2ᵌ ∙ 2⁷ = (2 x 2 x 2) (2 x 2 x 2 x 2 x 2 x 2 x2)
 2ᵌ +2 ⁷ =
 Answer = 29 = 512
 28
 Answer = 256
 You Try: 4ᵌ ∙ 4⁴
 Answer = 47 = 16, 384
 Another example: 6⁵ ∙ 6
 Answer = 66 = 66,656
MULTIPLICATION PROPERTIES
PRODUCT OF POWERS
This property is used to combine 2 or more exponential expressions with the SAME base.
2 2
3
3
4
( x )( x )
5
(2  2  2)(2  2  2  2  2)
( x)( x)( x) ( x)( x)( x)( x)
28
x7
256
MULTIPLICATION PROPERTIES
POWER TO A POWER
This property is used to write and exponential expression as a single power of the base.
(52 )3
(x 2 ) 4
(52 )(52 )(52 )
( x 2 )( x 2 )( x 2 )( x 2 )
56
x8
MULTIPLICATION PROPERTIES
POWER OF PRODUCT
This property combines the first 2 multiplication properties to simplify exponential expressions.
(6  5) 2
(5 xy )
3
(6) 2  (52 )
3
3
3
(5 )( x )( y )
(4 x 2 )3  x 5
(64)( x 6 )  x5
(43 )( x 2 )3  x5
64x11
36  25  900
125 x 3 y 3


(64) ( x 2 )( x 2 )( x 2 )  x5
MULTIPLICATION PROPERTIES
SUMMARY
PRODUCT OF POWERS
x x  x
a
b
a b
ADD THE EXPONENTS
POWER TO A POWER
x 
a b
x
ab
MULTIPLY THE EXPONENTS
POWER OF PRODUCT
( xy )  x y
a
a
a
Dividing Exponents
DIVISION PROPERTIES
QUOTIENT OF POWERS
This property is used when dividing two or more exponential expressions with the same base.
x 5 ( x)( x)( x)( x)( x) ( x)( x)
2


x
3
x
( x)( x)( x)
1
Division of Exponents Examples
•
54
5
 5*5*5*5
5
 Simplify: the numerator has 53 and nothing left in the
denominator.
 53 = 125
• 3
3⁴
 3
3 *3*3*3
 Simplify: the numerator has nothing left and the
denominator has 33 .
 Answer =
1
27
Negative Exponents
Negative Exponents
Negative Exponents
DIVISION PROPERTIES
4
x 
( x 2 ) 4 x8
 3   3 4  12
(y )
y
y 
2
Hard Example
3
3 3 6
2 3
3 12
 2 xy 
2 x y
(
2
xy
)
8
x
y
 3  4  
 3 9 12 

3 4 3
9 6
3 x y
(3x y )
27 x y
 3x y 
2
8 x 3 y12
8 y6

9 6
27 x 6
27 x y