Exponent Product Rule

Simple over of the exponent product/multiplication rule.

Exponent Product Rule Formula

When two factors have the same base and are raised to powers, we can rewrite their product as the single common base raised to the sum of the two powers.

\[a^m\cdot a^n=a^{m+n}\]

Watch this section: introduction at 0:00.

Why The Exponents Add

Consider the following specific example

\[x^2\cdot x^4\]

We can rewrite this as

\[\begin{align} x^2\cdot x^4 &= (x\cdot x)\cdot (x\cdot x\cdot x\cdot x) \quad \text{Writing out the explicit repeated base form of each factor.}\\ &= x\cdot x\cdot x\cdot x\cdot x\cdot x \quad \text{Because multiplication is associative.}\\ &= x^6 \end{align}\]

So essentially

\[x^2\cdot x^4 = x^{2+4}=x^6\]

Now, we can do same thing for two arbitrary powers $m$ and $n$.

\[x^m\cdot x^n = x^{m+n}\]

Now, instead of writing everything out in explicit form, then reassociating and combining again, we can just use the product rule to save time.

Watch this section: example of the property at 0:14.

Simple Example Using Exponent Product Rule

Consider

\[x^{10}x^6\]

Both factors have the same base, $x$, so the product rule applies immediately:

\[x^{10}x^6=x^{10+6}\]

Then add the exponents:

\[x^{10+6}=x^{16}\]

NOTE: If the base is not the same, we cannot apply the rule!

Watch this section: example 1 at 0:49.

Slightly More Confusing Example Using Exponent Product Rule

The second example has more than one base:

\[x^3y^2x^9y^{12}\]

We can reorder the product of our factors since multiplication is commutative.

\[x^3y^2x^9y^{12}=x^3x^9y^2y^{12}\]

Using the exponent product rule on the powers of $x$

\[x^3x^9=x^{3+9}=x^{12}\]

Using the exponent product rule on the powers of $y$

\[y^2y^{12}=y^{2+12}=y^{14}\]

So the simplified expression is

\[x^3y^2x^9y^{12} = x^{12}y^{14}\]

Be sure to match the base before applying the rule.

Watch this section: example 2 at 1:07.

Timestamp Guide

Section	What is shown	Video
Introduction	Set up when to add exponents.	0:00
Example of property	Introduce the product rule for matching bases.	0:14
Example 1	Combine $x^{10}x^6$ into one power.	0:49
Example 2	Combine $x^3y^2x^9y^{12}$ by matching bases.	1:07
Word of caution	Avoid adding exponents when bases do not match.	1:49

More Practice

For the timestamped list of product-rule examples already on the site, use the Exponent Product Rule Example Archive.

You can also review the broader exponents topic page or continue to the quotient rule for exponents.

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