What Is the Product Rule for Exponents?

By Amy Harris; Updated April 24, 2017
In Albert Einstein's famed equation,

The product rule for exponents states that to multiply powers with the same base add their exponents. The term “powers” refers to numbers written in exponential form, such as 12^3. “Base” refers to the number that is being raised to the power, and it appears directly to the left of exponent. For instance, in 12^5, the 12 is the base, and the 5 is the exponent, or power.

Expressed Arithmetically

The product rule is best illustrated by example. Suppose you’re multiplying 2^3 * 2^4, where the “*” symbol denotes multiplication. Keep the base, 2, as-is, and add the exponents: 4 + 3 = 7. This produces an overall solution of 2^7, which is equivalent to 2 * 2 * 2 * 2 * 2 * 2 * 2, or 128.

Applicability to Algebra

Although the product rule may initially be taught arithmetically, most of its usefulness pertains to algebra. Its technical definition is expressed algebraically: x^a * x^b = x^(a+b), where “a” and “b” symbolize integers, and “x” represents a number or variable. For instance, if you’re asked to simplify y^5 * y^4, the result is y^(5+4), which yields y^9.

A Common Misconception

The product rule only applies when the bases are the same. For instance, the product rule cannot be used to solve the problem 6^3 * 8^2, because the bases differ. Similarly, it would not pertain to x^5 * y^5.

About the Author

Based in western New York, Amy Harris began writing for Demand Media and Great Lakes Brewing News in 2010. Harris holds a Bachelor of Science in Mathematics from Penn State University; she taught high school math for several years and has also worked in the field of instructional design.