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.
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.