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.

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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

$\frac{1}{2}$

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Let’s look at an example with a variable.

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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

$\frac{1}{x^5}$

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Let’s look at another example.

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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 $b^7$.

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

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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

$3x^{5}$

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