Division and Reciprocals of Radical Expressions

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  • 0:03 Reciprocal of a Radical
  • 1:28 Division by a Radical
  • 2:42 Simplifying Radicals
  • 4:12 One More Example
  • 5:00 Lesson Summary
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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.

Reciprocal of a Radical

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.

Division by a Radical

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.

Simplifying Radicals

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.

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