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 Depdendent Variable

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 Dependent Variable

 Number of inequalities to solve: 23456789
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# Changing the Sign of an Exponent

Because a-n and an 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: a3 Â· a2 = a5

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

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