Changing the Sign of an Exponent
Because a^{n }and a^{n} are reciprocals of each other, we know that
So a negative exponent in the numerator or denominator can be changed to positive
by relocating the exponential expression. In the next example we use these facts to
remove negative exponents from exponential expressions.
Example 1
Simplifying expressions with negative exponents
Write each expression without negative exponents and simplify. All variables
represent nonzero real numbers.
Solution


Rewrite division as multiplication. 


Change the signs of the negative exponents. 


Product rule: a^{3} Â· a^{2} = a^{5}

Note that in 5a^{3} the negative exponent applies only to a.


Rewrite as multiplication. 


Definition of negative exponent 


Simplify. 
In Example 1 we showed more steps than are necessary. For instance, in part (b)
we could simply write
Exponential expressions (that are factors) can be moved from numerator to denominator
(or vice versa) as long as we change the sign of the exponent.
Caution
If an exponential expression is not a factor, you cannot move
it from numerator to denominator (or vice versa). For example,
Because
and 1^{1}
= 1, we get
