There are certain rules that govern working with exponential expressions. This lesson deals with the rule for multiplying exponential expressions. It will describe the rule and give some examples of how it works.

## What Is an Exponent?

An exponent is a number written as a superscript to another number.

It looks like this:

The **exponent** of a number tells you how many times to multiply that number to itself. So, in the above example, 2^3 means 2*2*2, which is equal to 8.Using the **carat** (^) symbol is another way to write an exponent that can be easier when typing.Exponents make large multiplication problems easier to write. So instead of writing 9*9*9*9*9*9, you can just write: 9^6.

## To Multiply, You Must Add

There are certain laws that govern working with exponents. One of these rules has to do with multiplying expressions that contain exponents.When you have two exponential expressions that have the same base, you can easily multiply them together. All you have to do is add the exponents.Here’s an example:(3^2)*(3^5)To simplify this expression, just add the exponents:2 + 5 = 7, and your answer is:3^7Let’s take a look at how this works.If we write out the multiplication of each exponent, we get(3*3)*(3*3*3*3*3), which equals:3*3*3*3*3*3*3To write this with exponents, we just count up the number of threes – there are 7 of them, so the simplified answer is 3^7.

This simplification works with all exponential expressions where the base is the same for each term.Let’s try another oneSimplify: (*x*^3)*(*x*^6)To simplify this expression, just add the exponents:3 + 6 = 9, so (*x*^3)*(*x*^6) = *x*^9The rule also applies if one or more of the exponents are negative.Simplify (4^-3)*(4^5)Again, just add the exponents.

-3 + 5 = 2So, (4^-3)*(4^5) = 4^2

## What if the Terms Have Different Bases?

If the terms you are working with have different bases, there is not much you can do to simplify the expression.For example:Simplify: (3^2)*(*x*^4)Since the bases for each term (3 and *x*) are different, nothing can be done to simplify this expression, and you are left with (3^2)(*x*^4).Simplify: (*b*^5)(*c*^3)The same rule applies to this example. Because the bases are not the same, nothing can be done to simplify the expression. The answer is (*b*^5)(*c*^3).

The only exception to this rule is if both the bases are numbers. Then, to simplify, you can simplify each term and multiply them together.For example, Simplify (2^3)*(6^2)2^3 = 8 and 6^2 = 36You can simplify this problem by multiplying 8 and 36.8 * 36 = 288

## Lesson Summary

Exponents are numbers written as superscripts that tell you how many times to multiply the base number to itself. In order to multiply two exponential expressions together, they must have the same base, and all you need to do is add the exponents. This works if the exponents are positive or negative, but only if the bases are the same.

If the terms have different bases, there is not much that can be done to simplify.

## Learning Outcomes

At the end of this lesson you should be able to:

- Understand and use the (^) symbol
- Simplify multiplication equations using exponents
- Simplify exponential equations with the same bases