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Chapter 5, Section 1
Exponents
We will use mainly rational numbers for bases and integers for exponents in this class –
in College Algebra and higher level courses you will begin using fractions as exponents.
Please note that fractional exponents follow the same rules as integer exponents.
Conventionally, exponents are reported as whole number whenever possible.
Rule one
The Product Rule
If factors have the same base, you may simplify by adding the exponents.
x s x t  x st
example:
2 2  23  4(8)  32  25
Practice Problems:
x 5 (x 3 ) 
2x 5 (3x 2 ) 
Rule Two
The Quotient Rule
If a factor in the numerator and a factor in the denominator have the same base, you may
subtract the exponents – you must subtract the exponent in the denominator from the
exponent in the numberator.
xs
 x st
t
x
Some facts that come about through the quotient rule:
x0  1
x 1 
1
x
Examples
57
53
x 11
x7
75
75
x3
x4
x 3 (x 7 )
x 2
32 x 5
31 x 2
Rule 3
The Power Rules
3A
( xy ) p  x p y p
(3x) 3  33 x 3  27x 3
example
3B
( x s ) t  x st
( x 2 ) 5  x 10
example
p
3C
x
xp
   p
y
y
3
example
Problems
3 1
5 2
2x 1
(2x ) 1
ab 2 (ab) 3
abc 
2 2
ab1
x (2x ) 2 y 3
6x 1 y 5
x
 
y
1
8
2
  
27
3
report your answers with whole number exponents
 2 1
(2) 0
 20
22
3 1
 2 2 ( xy ) 5
4x 3
5x 3 y 2
(5xy ) 2
x 2 (2xy 2 ) 3
2 1 x 2 y 1
 x 2
 3
y



2
(2x 1 ) 2 (2x ) 2
 (2x ) 2 (3y) 1
(3x 2 y 1 ) 2

(32 y) 1
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