Product of Square Roots Rule: Definition & Example

An error occurred trying to load this video.

Try refreshing the page, or contact customer support.

Coming up next: Addition and Subtraction Using Radical Notation

You're on a roll. Keep up the good work!

Take Quiz Watch Next Lesson
Your next lesson will play in 10 seconds
  • 0:04 Application of Square Roots
  • 0:53 Product of Square Roots Rule
  • 3:01 Some Examples
  • 4:24 Lesson Summary
Save Save Save

Want to watch this again later?

Log in or sign up to add this lesson to a Custom Course.

Log in or Sign up

Speed Speed

Recommended Lessons and Courses for You

Lesson Transcript
Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

Square roots show up often in the world around us. In this lesson, we'll go over the product of square roots rule. We'll look at multiple ways to apply this rule through various examples and explanations.

Application of Square Roots

Suppose you are trying to find out the length of a side of a square pen. You know that the area is 144ft 2, and you know that to find the area of a square, you would multiply the length of the side by itself. That is, if we let s be the length of a side of the square pen, we have that s 2 = 144. To solve this for s, we would take the square root of both sides and add in the plus or minus sign as shown:


Since we're talking about a length, and we can't have a negative length, we can discard the negative answer, so we have that s = sqrt(144).

You may be familiar with the fact that the square root of 144 is 12, but what do you do if you aren't familiar with this fact? How can you find the square root of 144 if you don't already know it? Well, let's find out one way that we can go about this!

Product of Square Roots Rule

We are going to look at a rule involving square roots that will allow us to break down a square root so that we can evaluate it more easily. That rule is called the product of square roots rule, and it states that the square root of a product is equal to the square root of each factor of the product multiplied together, or as you can see:


The rule itself may seem like common sense to you, but you may be left wondering how we can use this to help us find the square root of 144. Here's the trick - if you break down 144 into factors of perfect squares that you know, we can use the rule to break up the square root into a product of square roots and evaluate it this way. Let's see how this works.

Notice that 144 = 4 * 36, and 4 and 36 are well known square roots. Specifically, sqrt(4) = 2 and sqrt(36) = 6. Thus, we can use the product of square roots rule to break down and evaluate the square root of 144 as follows:


We see that the square root of 144 is 12, as expected. Now, you might be wondering what would happen if you had chosen different factors. For instance, suppose you factored 144 even further and got 144 = 4 * 36 = 4 * 4 * 9. Well, the product rule still applies, so let's see what happens when we do it this way:


Once again, we get that the square root of 144 is 12.

We see that this product of square roots rule is extremely handy when trying to evaluate large square roots. We can also use it to simplify square root expressions containing variables and to simplify square roots in general. Let's look at a few examples of this.

Some Examples

Suppose you wanted to simplify the following expression:


To unlock this lesson you must be a Member.
Create your account

Register to view this lesson

Are you a student or a teacher?

Unlock Your Education

See for yourself why 30 million people use

Become a member and start learning now.
Become a Member  Back
What teachers are saying about
Try it risk-free for 30 days

Earning College Credit

Did you know… We have over 200 college courses that prepare you to earn credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Transferring credit to the school of your choice

Not sure what college you want to attend yet? has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

Create an account to start this course today
Try it risk-free for 30 days!
Create an account