Exponent Power Rule

Simple overview of the exponent power rule.

Main Idea

We use the exponent power rule when a power is raised to another power.

\[\left(a^m\right)^n=a^{mn}\]

When a base is raised to a power, then raised to another power, you can simplify by replacing this with the base raise to the product of the two powers.

Watch this section: introduction at 0:00.

Why The Exponents Multiply

Consider

\[\left(x^n\right)^m\]

We can think of this as $x^n$ multiplied together $m$ times.

\[\begin{align} \left(x^n\right)^m &= (x^n)(x^n)\cdots(x^n) \quad \text{We have }m\text{ factors of }x^n \text{ all multiplied together.}\\ &= x^{n+n+\cdots + n} \quad \text{Adding }n \text{ together }m \text{ times.} \\ &=x^{m\cdot n} \quad \text{Repeated addition can be written as multiplication.} \end{align}\]

Instead of writing it out this way everytime, we can use the shorthand rule to quickly jump between forms.

\[\left(a^m\right)^n=a^{mn}\]

Watch this section: example showcasing the rule at 0:14.

Simple Example Applying the Rule

Consider the following

\[\left(x^5\right)^6.\]

The base is $x$. The inside exponent is $5$, and the outside exponent is $6$, so we multiply the powers:

\[\left(x^5\right)^6=x^{5\cdot 6}.\]

Then simplify:

\[x^{5\cdot 6}=x^{30}.\]

Watch this section: example 1 at 1:07.

Example with Multiple Factors Inside Parentheses

What happens when more than one factor exists within the parentheses?

Consider the following:

\[\left(x^5y^2z^3\right)^4.\]

The outside exponent applies to each factor in the product:

\[\left(x^5y^2z^3\right)^4 = \left(x^5\right)^4\left(y^2\right)^4\left(z^3\right)^4.\]

Now use the power rule on each base:

\[\left(x^5\right)^4=x^{20}, \qquad \left(y^2\right)^4=y^8, \qquad \left(z^3\right)^4=z^{12}.\]

So the expression becomes

\[\left(x^5y^2z^3\right)^4=x^{20}y^8z^{12}.\]

Watch this section: example 2 at 1:31.

Timestamp Guide

Section	What is shown	Video
Introduction	Set up the power-rule situation for exponents.	0:00
Why powers multiply	Expand a power of a power to show why the exponents multiply.	0:14
Example 1	Rewrite $\left(x^5\right)^6$ as one power.	1:07
Example 2	Rewrite $\left(x^5y^2z^3\right)^4$ by sending the outside power to each factor.	1:31
Warning	Do not treat sums inside parentheses like products.	2:15

More Practice

For timestamped practice from this video and related JoeCMath exponent lessons, use the Exponent Power Rule Example Archive.

You can also review the exponents topic page, compare this lesson with the exponent product rule companion, or continue to the exponent quotient rule companion.

The video description points to JoeCMath’s exponent property playlist: watch the playlist on YouTube.