## Product of Powers Property

## Introduction

An exponent is a mathematical operation that can be used for repeated multiplication. The number of the exponent tells you how many times to repeat the multiplication.6^{2} = 6 × 6 5^{3} = 5 × 5 × 5 2^{4} = 2 × 2 × 2 × 2

## Lessons

Each exponential expression has two parts: a base and an exponent (exponents can also be called powers). Exponents are a quick and efficient way to show a repeated multiplication.

The two examples above demonstrate how to evaluate an exponent. What do you do if you have several different exponential terms multiplied together? Consider the problem 5^{3} × 5^{4}.

In the problem above, the two expressions being multiplied both have a base of five. The work above shows that the exponents can simply be added. Here is another way to look at the work in similar problems.

The rules are the same whether the bases are numbers or variables. This property that allows you to multiply two exponential expressions with the same base is called the product of powers property.

This property works as long as the bases are the same in each of the exponential terms being multiplied.

Examples:

However, if the bases are different, then the product of powers property cannot be applied.

**More Complex Problems**

When dealing with problems that involve several different variables, multiply each variable separately and then combine the results to get your answer.

If you are even in doubt about how to do a problem that deals with exponents, consider expanding the exponents and then putting like terms together. For example, x^{2}∙x^{3} = x∙x ∙ x∙x∙x = x^{5}. Showing the work in this way may take a little extra time, but your extra time is worth it because you will be more confident in your answer and in your ability to do these types of problems.

## Review

Use the power of products property to multiply:

1) a^{2}∙a^{3}∙a^{4} =

2) x^{4}∙y^{2}∙z^{7} =

3) m^{2}n^{7 }∙ m^{3}n^{4 }=

4) 2^{2}s^{4}t^{6 }∙ 5s^{6}t^{11 }=

5) 9de^{5}f^{8 }∙ 2d^{3}e^{3}f^{5 }=

Scroll Down for Answers…

Use the power of products property to multiply:

1) a^{2}∙a^{3}∙a^{4} = a^{9}

2) x^{4}∙y^{2}∙z^{7} = x^{4}∙y^{2}∙z^{7}

3) m^{2}n^{7 }∙ m^{3}n^{4 }= m^{5}n^{11}

4) 2^{2}s^{4}t^{6 }∙ 5s^{6}t^{11 }= 20s^{10}t^{17}

5) 9de^{5}f^{8 }∙ 2d^{3}e^{3}f^{5 }= 18d^{4}e^{8}f^{13}

### Related Links:

Didn't find what you were looking for in this lesson? More information on exponents can be found at the following places:

**Resource Page**

**Related Algebra Lessons**

Looking for something else? Try the buttons to the left or type your topic into the search feature at the top of this page.