Negative exponents and power rule for exponents — Krista King Math

Negative exponents and power rule for exponents

How to deal with negative exponents

This lesson will cover how to find the power of a negative exponent by using the power rule.

Case 1 of the power rule for negative exponents:

If you have two positive real numbers ???a??? and ???b??? then

???b^{-a} = frac{1}{b^a}???

Think of it this way: in order to change the exponent in ???b^{-a}??? from ???-a??? to positive ???a??? you move the entire value from the numerator to the denominator to get

???frac{1}{b^a}???

Case 2 of the power rule for negative exponents:

If you have two positive real numbers ???a??? and ???b??? then

???frac{1}{b^{-a}}=b^a???

Think of it this way: in order to change the exponent in ???b^{-a}??? from ???-a??? to positive ???a??? you move the entire value from the denominator to the numerator to get ???1b^a??? which is the same as ???b^a???.

By the way, ???a^b??? and ???1/a^b??? are called reciprocals. Sometimes you’ll hear or read about negative exponents and their relationship to reciprocals and that’s because of this relationship.

Think about ???y^{-1}???. In order to change the exponent from ???-1??? to ???1??? you move the entire value from the numerator to the denominator to get

???frac{1}{y^1}???

???frac{1}{y}???

This means that ???y??? and ???y^{-1}??? are reciprocals.

How negative exponents and power rule for exponents are related to each other

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Rewriting expressions to eliminate the negative exponents

Example

Write the following without any negative exponents.

???2^{-1}???

In order to get rid of the negative exponent, we move the ???2^{-1}??? from the numerator to the denominator we get

???frac{1}{2^1}???

Which is the same as

???frac{1}{2}???

Let’s look at an example with a variable.

in order to change the exponent in b^(-a) from -a to positive a, you move the entire value from the denominator to the numerator to get 1b^a, which is the same as b^a.

Example

Get rid of the negative exponents.

???x^{-5}???

In order to get rid of the negative exponent, we move the ???x^{-5}??? from the numerator to the denominator. We get

???frac{1}{x^5}???

Let’s look at another example.

Example

Get rid of the negative exponents.

???frac{1}{b^{-7}}???

In order to get rid of the negative exponent, we move the ???b^{-7}??? from the denominator to the numerator. We get ???1b^7??? which is the same as ???b^7???.

Let’s look at a final example with a number other than ???1??? in the numerator.

Example

Write the expression without negative exponents.

???frac{3}{x^{-5}}???

In order to get rid of the negative exponent, we move the ???x^{-5}??? from the denominator to the numerator. We get

???3x^{5}???