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Algebra I: High School20 chapters | 168 lessons | 1 flashcard set

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Lesson Transcript

Instructor:
*Yuanxin (Amy) Yang Alcocer*

Amy has a master's degree in secondary education and has taught math at a public charter high school.

Watch this video lesson to learn what you need to do when dividing radical expressions. Learn what you need to look out for before you divide and why you can't leave your radical in your denominator.

In this video lesson, we are going to see what we need to do when we are dividing by radicals. There are actually two scenarios where we are dividing by a radical. The first is if we take the reciprocal of a radical. The second is when we are performing straight division with radicals. Keep watching to see what we need to do for each. You will also make use of your previous knowledge of simplifying radicals by finding their factors to see if there are any factors for which you can evaluate the radical. For example, the square root of 8 simplifies to 2 times the square root of 2. So, let's begin.

A **reciprocal of a radical** is the number 1 divided by your radical. So, if your radical happens to be the square root of 24, then your reciprocal is 1 divided by the square root of 24. Think of it as your radical trading sides with a 1. When this happens, you can sometimes evaluate right away if your radical is something you know. For example, if you take the reciprocal of the third root of 8, you can evaluate it right away because you know that the third root of 8 is 2, since 2 times 2 times 2 equals 8. But, if your radical is not something you can evaluate so easily, such as the square root of 24, then you will need to simplify your radical. We will be discussing simplifying your radical in just a bit. But before we do, we're going to discuss our second scenario of straight division of radicals.

We are going to go about dividing our radicals in two different ways depending on what kind of division problem we see. Recall that each radical has an index number which is the little number written in the little dip of the radical symbol. If our division problem has the same index for both the numerator and the denominator, and if the denominator divides evenly into the numerator, then we will go ahead and divide the numerator and denominator, combining them under the same radical symbol. So, for example, the third root of 24 divided by the third root of 4 becomes the third root of 24 divided by 4, which is the third root of 6. Now, if we had the third root of 4 divided by the third root of 8, we would actually go ahead and evaluate the third root of 8, because that would eliminate one of the radicals, thus simplifying our expression. So, we would have the third root of 4 divided by 2. So, a rule here is if we can go ahead and evaluate a radical thus removing the radical, then we should by all means do so. Now, if we have any other case where we still have a radical in the denominator, then we'll have to simplify our radical so that we don't have a radical in the denominator. This is what we will talk about now.

One of the rules of simplifying radicals is that we have to remove the radicals from the denominator. We can't have radicals in the denominator. So, if either our reciprocal of a radical or our division by a radical gives us a radical in the denominator that we can't evaluate, then we would have to use the method that I'm going to show you right now to remove it.

This method involves multiplying the numerator and denominator by the radical in the denominator. So, for 1 divided by the square root of 24, I would multiply the 1 with a square root of 24, and we would multiply the square root of 24 with the square root of 24. What happens when we multiply a radical by itself? We get the number inside the radical. So the square root of 24 multiplied by the square root of 24 gives us 24. So 1 divided by the square root of 24 simplifies into the square root of 24 divided by 24. Now, we aren't quite done yet. We still need to check our radicals to see if we can simplify them even further. The square root we can actually simplify further, because we can split it into the square root of 4 times the square root of 6 which becomes 2 times the square root of 6. Now we see that we have a 2 over a 24 so we can actually simplify this even further by dividing the 2 and the 24. So, our final answer is the square root of 6 over 12.

Let's look at one last example. Let's say we are dividing the square root of 11 by the square root of 3. I can't combine them under the same radical because 3 does not divide evenly into 11. I can't evaluate the square root of 3 in the denominator, either. So that means I need to simplify this division problem by multiplying both the numerator and denominator by my radical in the denominator - my square root of 3. So multiplying the numerator by the square root of 3 gives me the square root of 11 times the square root of 3, which becomes the square root of 33. The denominator is the square root of 3 multiplied by the square root of 3 which becomes 3. I can't simplify anything else any further, so my answer is the square root of 33 divided by 3.

What have we learned? We've learned that when it comes to dividing by a radical or having the **reciprocal of a radical**, a number 1 divided by your radical, we must strive to remove the radical in the denominator. If the indexes of the numerator radical and denominator radical are the same, we should see if the numbers divide evenly into each other. Or if we can evaluate any of the radicals, then we should do so as well. Our goal is to eliminate the radical in the denominator. If we can't easily remove the radical from the denominator, then we must simplify our radical by multiplying both our numerator and denominator by the radical in the denominator. Doing so removes the radical in the denominator leaving us with the value inside the radical. The numerator will be multiplied by that radical. This is all part of simplifying our radical.

View this video lesson as you strengthen your ability to:

- Identify the reciprocal of a radical
- Understand the process of dividing by a radical number
- Remember the steps necessary to simplify a radical expression

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Algebra I: High School20 chapters | 168 lessons | 1 flashcard set

- How to Find the Square Root of a Number 5:42
- Estimating Square Roots 5:10
- Simplifying Square Roots When not a Perfect Square 4:45
- Simplifying Expressions Containing Square Roots 7:03
- Division and Reciprocals of Radical Expressions 5:53
- Evaluating Square Roots of Perfect Squares 5:12
- Factoring Radical Expressions 4:45
- Simplifying Square Roots of Powers in Radical Expressions 3:51
- Multiplying then Simplifying Radical Expressions 3:57
- Dividing Radical Expressions 7:07
- Simplify Square Roots of Quotients 4:49
- Rationalizing Denominators in Radical Expressions 7:01
- Addition and Subtraction Using Radical Notation 3:08
- Multiplying Radical Expressions with Two or More Terms 6:35
- Solving Radical Equations: Steps and Examples 6:48
- Solving Radical Equations with Two Radical Terms 6:00
- Go to High School Algebra: Radical Expressions

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