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Rationalizing the Denominator with Higher Roots - Concept 17,502 views

Teacher/Instructor Carl Horowitz
Carl Horowitz

University of Michigan
Runs his own tutoring company

Carl taught upper-level math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!

When a denominator has a higher root, multiplying by the radicand will not remove the root. Instead, to rationalize the denominator we multiply by a number that will yield a new term that can come out of the root. For example, with a cube root multiply by a number that will give a cubic number such as 8, 27, or 64.

Rationalizing the denominator is basically a way of saying get the square root out of the bottom. Okay. We can ask why it's in the bottom. Not really sure why but but for some reason we can't and when we do it we need to multiply by something in order to get rid of the square root.
So we're going to do an example let's hope that you remember how to do it. And that's going to be 4 over square root of 8, okay? We could multiply it by the square root of 8 by the square root of 8, square root of 8 is then cancel leaving us with 8. But what I want to get you in the habit of doing is looking to see if there is a way that we can simplify the denominator first. Okay? And what I mean by that is basically make it so we have a smaller square root than we're dealing with. Square root of 8 is the same thing as square root of 4 times the square root of 2, which is just 2 times root 2. So this expression is the same thing as 4, 2 root 2. Okay? So simplify this up, and now we're only concerned with the square root of 2, that is in the denominator. People often want to multiply by the entire denominator. Don't, you don't have to do that. Root 2 is the only thing creating the problem so you can leave the 2 right there as it is. Okay?
So in order to rationalize the denominator multiply by root 2 over root 2. Our numerator becomes 4 root 2, our 2 is still there and then we have root 2 times root 2 which is just 2. You can simplify this up 4 over 2 times 2, they all cancel just leaving us with the square root of 2, okay. So hopefully that's nothing too too new for you.
What I want to talk about mostly now is the denominator when you're dealing with a root other than 2. So this example what we're talking about is the cube root. And one common mistake is people want to multiply by the cube root of 2 over cube root of 2. Okay? When we are multiplying radicals, we [IB] base the roots are the same. We combine this. So we actually end up here is the cube root of 2 squared or 4. That doesn't help us at all though because the cube root of 4 we don't know. So what you really need to think about is how many this of a term do you need in order to get it out of a cube root? Okay? In order to get something out of a cube root you need 3 of them, alright? The cube root of 2 cubed is 2, the cube root of 8 is 2. So in order to get something out of a cube root you need 3 items] and to get something out of a fourth root you need 4. So thinking about this, I need the cube root of 2 squared. I have one 2 I then need two more to make three. Okay? Multiplying the top and the bottom by the same thing the cube root of 2 squared. We now have the 3 times the cube root of 2 squared over the cube root of 2 to the third. Cube root of 2 to the third is just 2. So we end up with just 3, cube root of 4 over 2. That 3 doesn't look very good, let's rewrite that.
So whenever we are dealing with a root other than a square root, you need to really think about your index. Think about the root and make sure you have that number of an element in order to pull it out, okay? You can't just do the same approach as this one over here multiplied with the root of whatever. You really need to reflect on it, say okay. I have a fifth root therefore I'm going need to need 5 items for whatever is at at hand.