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Hawkes Learning Systems:
College Algebra
Section 1.3a: Properties of Exponents
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Objectives
o Natural number exponents.
o Integer exponents.
o Properties of exponents and their use.
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Natural Number Exponents
If a is any real number and if n is any natural number
then a n  a  a  . ..  a .
n facto rs
In the expression a n , a is called the base and n is the
exponent.
a
base
Note: a 1  a
n
exponent
Read “a to the n th power”
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Example 1: Natural Number Exponents
Expression Base(s) Exponent(s)
2
3
 4
2
2
4
   1  6
3
x x
2
5
7
5
4
5
2
Solution
2
3
222  8
4
2
  4     4   16
2
4
  2  2  2  2    16
 1, 6
3, 2
    1     1     1     6  6   36
x, x
2, 5
5, 5
7, 4
 x  x x  x  x  x  x 
5555555
5555
x
7
 5  5  5  125
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Integer Exponents
If n and m are natural numbers, then
a a
n
m
  a  a  ...  a    a  a  ...  a   a
n factors
n
m
n m
Thus, a  a  a
.
m factors
n m
.
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0 as an Exponent
0
m
0 m
m
 a.
If n=0, then a  a  a
This suggests the following:
For any real number a  0 , we define a 0  1.
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Integer Exponents
Consider the following:
2 8
3
The result is 1/2 of the
result from the previous
line. What, then, do you
know about the following
expressions with negative
exponents?
2  4
2
The exponent is decreased
by one at each step.
2  2
1
2 1
0
2
2
2
1
2
3
?

?

?

1
2
1
4
1
8
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Integer Exponents
Negative Integer Exponents
For any real number a  0 and for any natural number n,
a
n

1
a
n
( a  0 simply to avoid the possibility of division by 0.)
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Example 2: Integer Exponents
Simply the following.
a.
b.
y
3
y
5
4x

yyy
yyyyy
3
2 x
3

4
4
2
c. 7  7  7
0

1
y
2
 y
2
 2
0 4
7
4

1
2401
Continued on the next slide…
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Example 2: Integer Exponents (cont.)
d.
2
1
s
2
s
1
2
 1

 s
1
1
2
s
e.  x y
3
2

2
 x y
3
2
x
3
y
2

 x x y y  x y
3
3
2
2
6
4
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Properties of Exponents and Their Use
Properties of Exponents
Throughout this table, a and b may be taken to represent constants, variables
or more complicated algebraic expressions. The letters n and m represent
integers.
Property
Example
3
1
3  (  1)
2
1. a n  a m  a n  m
3 3  3
3 9
2.
a
n
a
m
3. a
n
 a
nm
7
7

1
a
n
5
9
10
2
 7

9 1 0
1
5
2

 7
1
25
1
and x 
3
1
x
3
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Properties of Exponents and Their Use
Property
m
n
4.  a   a nm
5.  ab   a b
n
n
Example
2 
3
n
2
7x
 2
3
n
n
6.  a   a
 
n
b
b
 
2
 2  64
6
 7 x  343 x and
 2 x 
5
3 2
3
2
3
  2
2
3
2
x 
5
2
 4x
10
2
2
3
9
1
1
3
 1 


and


 


2
2
2
2
x
x
x
3
z
9
z
 


3z 
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Example 3: Properties of Exponents and Their
Use
Simplify the following by using the properties of exponents.
Write the final answer with only positive exponents.
a. (23 x 4  6 x 3  2 x  13) 0 Any non-zero expression with an exponent of 0 is 1.
1
Continued on the next slide…
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Example 3: Properties of Exponents and Their
Use (cont.)
b.  x y
3
2
4

x z
2
z
3
1

x
6
y
4
4
x z

z
6
z
1
3
We could have used the same properties in a
different order to achieve the same result.
Ex:  x 3 y 2   2 z  1
3 2
4
x z
3


10
4
4

4
2
y
z
6

z
10

2
2
4
x y x
2
x y
x
3
3 2
x x y z
z
z
2
x y
4
4
4
x z
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Example 4: Properties of Exponents and Their
Use
Simplify the following by using the properties of exponents.
Write the final answer with only positive exponents.
 5 xz   3 x
3
3
9
3x y
3
2
125 x
3
3


3
5 x z
3
2
3x y z

125
2
3y z
9
9
3
y
2

1
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Properties of Exponents and Their Use
The first column contains example of common errors and
the second column contains the corrected statements.
Incorrect Statements
Correct Statements
x x  x
5
10
x x  x
5
7
2 2  4
7
2 2  2
7
3  4  7
2
2
4
3
3  4
x
2
2
1
 3y
3x 
x
x
2
4
3 4
1
1
 2 
x
3y
2
 3x
2
5
2
2
 x
3
3
2
2
x
2
 3y
1
3x 
x
x
2
1

x2  3y
 9x
2
5
2
 x
7