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RATIONAL EXPONENTS
Laws of
Exponents
Multiplication
Properties
•Product of powers
•Power to a power
•Power of a product
Zero and negative
exponents
Division properties
of exponents
•Quotient of powers
•Power of a quotient
Basic Terminology
Exponent  In Exponential notation, the number of times the base
is used as a factor.
Base  In Exponential notation, the number or Variable that undergoes repeated
MULTIPLICATION .
2
4
EXPONENT
means
2  2  2  2  16
BASE
Important Examples
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 Examples
Variable Expressions
x 5 means (use parentheses for multiplica tion) ( x)( x)( x)( x)( x)
y 3 means ( y)( y)( y)
Substitution and Evaluating
STEPS
1. Write out the original problem.
2. Show the substitution with parentheses.
3. Work out the problem.
Example : Solve if x  4;
Solve if x  4;
(4)
3
x3
x3
= 64
More Examples
Evaluate the variable expression when x = 1, y = 2, and w = -3
( x)  ( y )
2
2
( x  y)
Step 1
( x)  ( y )
2
2
( x  y)
Step 1
2
wx
y
Step 2
Step 2
(1)  (2) 
2
Step 3
1 4  5
wx
y
Step 1
Step 2
(1) 2  (2) 2
2
Step 3
(3)  9
2
(3)(1)
2
Step 3
(3)(1)  3
MULTIPLICATION PROPERTIES
PRODUCT OF POWERS
This property is used to combine 2 or more exponential expressions with the SAME base.
2 2
3
5
3
4
( x )( x )
(2  2  2)(2  2  2  2  2)
( x)( x)( x) ( x)( x)( x)( x)
28
256
x7
MULTIPLICATION PROPERTIES
POWER TO A POWER
This property is used to write and exponential expression as a single power of the base.
2 3
(5 )
(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 )  x 5
(43 )( x 2 )3  x 5
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
DISTRIBUTE THE EXPONENT
( xy)  x y
a
a
a
ZERO AND NEGATIVE EXPONENTS
ANYTHING TO THE ZERO POWER IS 1.
33  27
32  9
31  3
30  1
1 1
3  1
3 3
1 1
2
3  2 
3
9
1
1
33  3 
3
27
1
 1 2
2 x  2 2   2
x  x
1
1
1
2
(2 x) 
 2 2  2
2
(2 x)
2 x
4x
2
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 PROPERTIES
POWER OF A QUOTIENT
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
(3x y ) 3 x y
27 x y
 3x y 
2
8 x 3 y12
8 y6

9 6
27 x 6
27 x y
Summary of Zero, Negative, and
Division Properties
ZERO, NEGATIVE, AND DIVISION
PROPERTIES
Zero power
( x)  1
Negative Exponents
x
a
1
 a
x
and
1
a
x
a
x
0
Quotient of powers
xa
a b

x
b
x
Power of a quotient
a
x
x
   a
y
 y
a