Negative Integer Exponents
Definition of a Negative Integer Exponent
You have seen how to work with exponents that are positive integers or 0.
Now, we will investigate exponents that are negative integers.
Weâ€™ll begin with this expression:


We can simplify the expression by
canceling common factors.


We can also simplify the expression
by subtracting exponents. 

Since
simplifies to both
and 2^{3}, we conclude
.
This relationship holds in general.
Note:
Recall:
2^{3} = 2 Â· 2 Â· 2
2^{0} = 1
Definition â€” Negative Integer Exponent
Here, x ≠ 0 and n is a nonnegative integer.
Example 1
Find: 5^{2}
Solution
Use the definition of a negative exponent: 



So,
We can also define
as follows:
Definition â€”
Here, x ≠ 0 and n is a nonnegative integer.
Example 2
a) Find:
b) Find:
Solution
a. Use the definition


b. Use the definition


Note:
A negative exponent does not determine
if an expression is positive or negative.
For example:

positive 

negative 
