Product of Square Roots Rule: Definition & Example

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  • 0:04 Application of Square Roots
  • 0:53 Product of Square Roots Rule
  • 3:01 Some Examples
  • 4:24 Lesson Summary
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Lesson Transcript
Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

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:


sqrprod1


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:


sqrprod2


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:


sqrprod3


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:


sqrprod4


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:


sqrprod5


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