What to do with negative exponents

How do you deal with negative exponents?

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

Part 1: A reminder

Remember that any number can be written as itself divided by $1$. For example, $3$ is the same as $3/1$. Also remember that the top part of a fraction is called the numerator and the bottom part of a fraction is called the denominator.

Part 2: The rule for negative exponents

If you have two positive real numbers $a$ and $b$, then

$b^{-a} = \frac{1}{b^a}$

You can think of it like this: first we need to realize that $b^{-a}$ is the same as

$\frac{b^{-a}}{1}$

We’ll change the exponent in $b^{-a}$ from $-a$ to $a$ by moving the entire value from the numerator to the denominator to get the

$\frac{1}{b^a}$

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Let's take a brief timeout to talk about reciprocals

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 $4^{-1}$. First realize that $4^{-1}$ is the same as

$\frac{4^{-1}}{1}$

We’ll change the exponent in $4^{-1}$ from $-1$ to $1$ by moving the entire value from the numerator to the denominator to get

$\frac{1}{4^{1}}$

$\frac{1}{4}$

This is a video with lots of examples of how to change negative exponents to positive exponents and vice versa

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An example of changing a negative exponent into a positive exponent

Example

Simplify the expression.

$4^{-2}$

Remember that $4^{-2}$ is the same as

$\frac{4^{-2}}{1}$

We’ll change the exponent from $-2$ to $2$ by moving the entire value from the numerator to the denominator to get,

$\frac{1}{4^2}$

Now we’ll perform the calculation in the denominator.

$\frac{1}{4^2} = \frac{1}{4 \cdot 4}$ $= \frac{1}{16}$

Moving a term from the numerator to the denominator, or vice versa, changes the sign of the exponent

An example with a negative sign in front of the base

Example

Simplify the expression.

$-5^{-3}$

Remember, we can rewrite $-5^{-3}$ as

$\frac{-5^{-3}}{1}$

because they are the same value.

We’ll change the exponent from $-3$ to $3$ by moving the entire value from the numerator to the denominator.

$\frac{1}{-5^3}$

We have to apply the exponent before we apply the negative sign so the expression becomes

$\frac{1}{-125}$

$-\frac{1}{125}$

A negative exponent when the base is a variable

Example

Write the expression with only positive exponents.

$x^{-3}$

First, we need to realize that the expression $x^{-3}$ is the same as

$\frac{x^{-3}}{1}$

We’ll change the exponent from $-3$ to $3$ by moving the entire value from the numerator to the denominator.

$\frac{1}{x^3}$

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