# Square & Cube Roots of Monomials

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Lesson Transcript
Instructor: Kevin Newton

Kevin has edited encyclopedias, taught middle and high school history, and has a master's degree in Islamic law.

Being able to find the cube or square root of monomials can make all math much easier, and it's not as hard as you'd think. This lesson will show you how and allow you to test your skills with a quiz.

## Review of Square and Cube Roots

So, as you've probably guessed, this is a lesson on how to take the square root and cube root of a monomial. Remember, a monomial is simply a term that doesn't have any external operations. So, 3x^2 is a monomial, but not x + 5x^2. That's the easy part.

Now, on to taking the square root and cube root. A square root, as you may recall, is a number times itself which gives you your original number. Likewise, a cube root is a number that is multiplied by itself twice to arrive at the original number.

If you thought that I was going to go over that the cube root of 27 is 3 or that the square root of 49 is 7, then you're going to get more than you bargained for. Yes, those are the right answers. But what is the square root of 98?

If you guessed that you break it into the square root of 49 and the square root of 2, then solve, leaving 7 times the square root of 2, then you'd be on the right path. Even better, if you got that the cube root of 54 was really the cube root of 2 times the cube root of 27, so it's 3 times the cube root of 2, then you've really got it. If not, just look at how I broke the number into smaller parts; that is going to be important here in a second.

## Square Root of a Monomial

First, let's start with square roots. Let's say that you go from taking the square root of friendly numbers like 49, 64, or 100, and suddenly end up with a weird-looking number, like the square root of 16 * x^4. What do you do? Same as before - break it apart. The square root of 16 is easy enough, but what about x^4?

Here's the trick for finding the square root of a variable, especially a variable with an exponent. Simply divide the exponent by 2. After all, if you were to think of it a little differently, the square root of a number is really just that number to the 1/2 power. So, if you divide the 4 in x^4 by 2, you get x^2. Therefore, the answer is 4x^2.

Before we go on, let's make sure that you've got this. Say you wanted to take the square root of 64 * a^6 * b^8. Wait, I didn't say there could be two variables! However, the process is just the same, except this time, let's break it into three parts.

What is the square root of 64? That's easy; it's 8 since 8 times 8 is 64. Now for the variable parts. Remember just to divide the exponent by 2. Therefore, you end up with 6 / 2, or a^3, and 8 / 2, or b^4. Therefore, your final answer is 8 * a^3 * b^4.

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