Product Theorem for Exponents: Definition & Examples

Instructor: Stephanie Matalone

Stephanie taught high school science and math and has a Master's Degree in Secondary Education.

In this lesson, we will go over the basics of the Product Theorem for Exponents, explaining how to multiply terms with exponents in them. We will discuss different examples and how to solve them.

That is a Lot of Zeros

There are 100,000 guests at a party your company is holding. You are in charge of the party favors and your boss demands that each guest receives a gift card with $100 on it. You need to quickly figure out how much money worth of gift cards to give out but you do not have a calculator or time to write it all out. Not to fret! We can use exponents to simplify your problem and solve things easily.

100,000 can be written as the exponent 105. This is because 10 multiplied by itself 5 times will result in 100,000. 100 can be written as the exponent 102. To figure out how much money worth of gift cards you need, you will simply multiply 105 by 102. In order to multiply these two exponential terms with a base of ten, we must use the Product Theorem for Exponents.

Product Theorem for Exponents

Before we get ahead of ourselves, let's review some definitions. The Product Theorem for Exponents is a rule that governs how we multiply exponential terms with the same base. The base is the number to the left in an exponential term. It is the number being multiplied. The number on the top right is called a power or an exponent and tells you how many times to multiply the base by itself. In the term 105, 10 is the base and 5 is the exponent. This tells us to multiply ten by itself five times.

The Product Theorem for Exponents looks like this:

Product Theorem for Exponents

The Product Theorem of Exponents states that when two exponential terms that have the same base are multiplied together, the exponents (powers) are added. Thus, when we multiply 105 by 102, we will add the two exponents (5+2) to get 7. The base of ten stays as it is. This gives us 107 as our answer. 107 is equal to 10,000,000. This means you will be giving out $10,000,000 in gift cards! Your boss might want to reconsider his generosity.

multiplying ten

Example with Variables

Let's try some with variables, the unknowns which are represented by letters, which you will often see in your math classes. Let's say you need to multiply the x4 by x6. Since both terms have a base of x, we can use the product theorem again and add the exponents to get x10.

multiplying variables

Example with Multiple Variables

Not all problems will be that straightforward. Sometimes you will be given a problem with more than one variable. Let's say you are asked to multiply x2y9z by y8z3x. Do not get overwhelmed by all the variables and numbers, just add the exponents together for the terms with the same base. This will give you an answer of x3y17z4. In our answer, we always put variables in alphabetical order which is why x is first in the answer and z is last.


Note that when a variable does not look like it has an exponent, it actually has an exponent of 1 (x = x1).

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