Product Of Powers: Definition, Rule & Property

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  • 0:01 Product of Powers Rule
  • 0:11 Definition
  • 1:48 Examples
  • 4:50 Lesson Summary
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Lesson Transcript
Instructor: Jennifer Beddoe

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

The product of powers rule states that when multiplying two powers with the same base, just add the exponents. Confused yet? This lesson will clear everything up, give some examples and provide a quiz for you to test your knowledge.

Product of Powers Rule


The product of powers rule tells us that when you are multiplying two terms that have the same base, you can just add their exponents to find your answer.

Not sure what that means? Let's break it down.


The product of powers rule assists with simplifying exponents. Let's first define some terms as they relate to exponents. When you have a number or variable raised to a power, the number (or variable) is called the base, while the superscript number is called the exponent, or power.


In a multiplication problem involving exponents, if the bases are the same, the multiplication problem can be simplified by adding the exponents.

For example, look at this problem: m^3 * m^5 = ?

Because the base (m) is the same, you can simplify the problem by adding the exponents: 3 + 5 = 8. Therefore, m^3 * m^5 = m^8.

Here is how this works:

  • m^3 = mmm (3 ms multiplied together)
  • m^5 = mmmmm (5 ms multiplied together)

Since the problem now asks us to multiply these two items together, we end up with the following: m^3 * m^5 = mmmmmmmm, which is equal to m^8 (or 8 ms multiplied together). Too bad they aren't M&Ms!


You can use the product of powers rule for simple or more complex multiplication problems. Let's apply it together.

1. Simplify f^2 * f^7

Since the bases are the same, to solve this problem, we just add the exponents. Pause the video to try this before you see the answer onscreen.

Did you get f^2 * f^7 = f^9? Great! Let's do a more complicated problem.

2. Simplify a * b^5 * c^2 * a^3 * b^2 *c^3

Remember that you can only combine like terms. Those are the terms with the same bases. If it helps you to rearrange the problems so that terms with like bases are together, no problem. Just don't leave anything out. Since this is a multiplication problem, there is not as much to keep track of as there would be with addition or subtraction or even division.

Consider pausing the video to try this yourself before the answer is revealed for each step so you can check your progress.

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