Power Of A Product: Property & Rule

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  • 0:01 Power of a Product…
  • 1:10 How Does It Work?
  • 1:50 Examples
  • 3:23 Lesson Summary
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Lesson Transcript
Instructor: Jennifer Beddoe

Jennifer has an MS in Chemistry and a BS in Biological Sciences.

The Power of a Product rule states that a term raised to a power is equal to the product of its factors raised to the same power. In this lesson, learn more about this rule and look at some examples.

Power Of A Product Rule Defined

The Power of a Product rule is another way to simplify exponents. First, we need to define some terms as they relate to exponents. When you have a number or variable raised to a power, it is called the base, while the superscript number, or the number after the '^' mark, is called the exponent or power.

If there is more than one term in parenthesis, with an exponent outside the parenthesis, then the exponent is distributed to every term in the parenthesis. For example, consider this problem:


Because of the parenthesis, the exponent (5) should be distributed to each term.


(mn)^5 = (m^5)(n^5)

There are two conditions that must be met in order for the Power of a Product rule to work.

  • First, there must be two or more variables or constants that are being multiplied together. In the example we just looked at, those are the m and n, but they could be any variable or constant.
  • Second, the result of the multiplication problem must be raised to a power. In the example, that is the 5.

Both conditions must be present in order to use the Power of a Product rule.

How Does It Work?

The Power of a Product rule can be proven by testing it using only numbers.

(4 * 2)^3

Using the Power of a Product rule, the solution is:

4^3 * 2^3 = 64 * 8 = 512

Then, work the problem like a simple math problem.

(4 * 2)^3 = 8^3 = 512

No matter what two numbers and exponent you use, the answer reached mathematically will always equal the answer found when you use the Power of a Product rule to solve it. Therefore, using the Power of a Product rule will also work when the problem contains variables.


You can use the Power of a Product rule for simple or more complex problems.

Example 1

Simplify: (ab)^7

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