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In this video, we're going to learn how to rationalize the

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denominator.

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What we mean by that is, let's say we have a fraction that

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has a non-rational denominator, the simplest one

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I can think of is 1 over the square root of 2.

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So to rationalize this denominator, we're going to

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just re-represent this number in some way that does not have

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an irrational number in the denominator.

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Now the first question you might ask is,

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Sal, why do we care?

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Why must we rationalize denominators?

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And you don't have to rationalize them.

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But I think the reason why this is in many algebra

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classes and why many teachers want you to, is it gets the

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numbers into a common format.

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And I also think, I've been told that back in the day

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before we had calculators that some forms of computation,

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people found it easier to have a rational number in the

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denominator.

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I don't know if that's true or not.

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And then, the other reason is just for aesthetics.

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Some people say, I don't like saying what 1

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square root of 2 is.

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I don't even know.

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You know, I want to how big the pie is.

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I want a denominator to be a rational number.

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So with that said, let's learn how to rationalize it.

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So the simple way, if you just have a simple irrational

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number in the denominator just like that, you can just

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multiply the numerator and the denominator by that irrational

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number over that irrational number.

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Now this is clearly just 1.

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Anything over anything or anything over that same thing

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is going to be 1.

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So we're not fundamentally changing the number.

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We're just changing how we represent it.

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So what's this going to be equal to?

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The numerator is going to be 1 times the square root of 2,

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which is the square root of 2.

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The denominator is going to be the square root of 2 times the

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square root of 2.

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Well the square root of 2 times the square

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root of 2 is 2.

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That is 2.

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By definition, this squared must be equal to 2.

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And we are squaring it.

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We're multiplying it by itself.

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So that is equal to 2.

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We have rationalized the denominator.

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We haven't gotten rid of the radical sign, but we've

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brought it to the numerator.

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And now in the denominator we have a rational number.

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And you could say, hey, now I have square root of 2 halves.

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It's easier to say even, so maybe that's another

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justification for rationalizing this

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denominator.

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Let's do a couple more examples.

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Let's say I had 7 over the square root of 15.

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So the first thing I'd want to do is just simplify this

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radical right here.

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Let's see.

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Square root of 15.

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15 is 3 times 5.

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Neither of those are perfect squares.

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So actually, this is about as simple as I'm going to get.

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So just like we did here, let's multiply this times the

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square root of 15 over the square root of 15.

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And so this is going to be equal to 7 times the square

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root of 15.

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Just multiply the numerators.

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Over square root of 15 times the square root of 15.

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That's 15.

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So once again, we have rationalized the denominator.

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This is now a rational number.

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We essentially got the radical up on the top or we got the

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irrational number up on the numerator.

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We haven't changed the number, we just changed how we are

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representing it.

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Now, let's take it up one more level.

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What happens if we have something like 12 over 2 minus

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the square root of 5?

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So in this situation, we actually have a binomial in

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the denominator.

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And this binomial contains an irrational number.

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I can't do the trick here.

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If I multiplied this by square root of 5 over square root of

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5, I'm still going to have an irrational denominator.

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Let me just show you.

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Just to show you it won't work.

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If I multiplied this square root of 5 over square root of

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5, the numerator is going to be 12 times the

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square root of 5.

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The denominator, we have to distribute this.

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It's going to be 2 times the square root of 5 minus the

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square root of 5 times the square root of 5, which is 5.

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So you see, in this situation, it didn't help us.

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Because the square root of 5, although this part became

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rational,it became a 5, this part became irrational.

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2 times the square root of 5.

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So this is not what you want to do where you have a

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binomial that contains an irrational number in the

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denominator.

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What you do here is use our skills when it comes to

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difference of squares.

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So let's just take a little side here.

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We learned a long time ago-- well, not that long ago.

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If you had 2 minus the square root of 5 and you multiply

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that by 2 plus the square root of 5, what will this get you?

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Now you might remember.

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And if you don't recognize this immediately, this is the

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exact same pattern as a minus b times a plus b.

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Which we've seen several videos ago is a

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squared minus b squared.

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Little bit of review.

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This is a times a, which is a squared. a

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times b, which is ab.

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Minus b times a, which is minus ab.

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And then, negative b times a positive

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b, negative b squared.

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These cancel out and you're just left with a

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squared minus b squared.

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So 2 minus the square root of 5 times 2 plus the square root

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of 5 is going to be equal to 2 squared, which is 4.

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Let me write it that way.

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It's going to be equal to 2 squared minus the square root

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of 5 squared, which is just 5.

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So this would just be equal to 4 minus 5 or negative 1.

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If you take advantage of the difference of squares of

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binomials, or the factoring difference of squares, however

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you want to view it, then you can rationalize this

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denominator.

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So let's do that.

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Let me rewrite the problem.

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12 over 2 minus the square root of 5.

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In this situation, I just multiply the numerator and the

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denominator by 2 plus the square root of 5 over 2 plus

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the square root of 5.

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Once again, I'm just multiplying the number by 1.

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So I'm not changing the fundamental number.

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I'm just changing how we represent it.

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So the numerator is going to become 12

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times 2, which is 24.

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Plus 12 times the square root of 5.

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Once again, this is like a factored

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difference of squares.

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This is going to be equal to 2 squared, which is going to be

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exactly equal to that.

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Which is 4 minus 1, or we could just-- sorry.

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4 minus 5.

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It's 2 squared minus square root of 5 squared.

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So it's 4 minus 5.

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Or we could just write that as minus 1, or negative 1.

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Or we could put a 1 there and put a

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negative sign out in front.

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And then, no point in even putting a 1 in the

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denominator.

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We could just say that this is equal to negative 24 minus 12

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square roots of 5.

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So this case, it kind of did simplify it as well.

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It wasn't just for the sake of rationalizing it.

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It actually made it look a little bit better.

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And you know, I don't if I mentioned in the beginning,

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this is good because it's not obvious.

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If you and I are both trying to build a rocket and you get

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this as your answer and I get this as my answer, this isn't

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obvious, at least to me just by looking at it, that they're

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the same number.

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But if we agree to always rationalize our denominators,

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we're like, oh great.

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We got the same number.

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Now we're ready to send our rocket to Mars.

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Let's do one more of this, one more of these right here.

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Let's do one with variables in it.

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So let's say we have 5y over 2 times the square

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root of y minus 5.

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So we're going to do this exact same process.

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We have a binomial with an irrational denominator.

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It might be a rational.

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We don't know what y is.

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But y can take on any value, so at points it's going to be

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irrational.

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So we really just don't want a radical in the denominator.

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So what is this going to be equal to?

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Well, let's just multiply the numerator and the denominator

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by 2 square roots of y plus 5 over 2 square

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roots of y plus 5.

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This is just 1.

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We are not changing the number, we're just

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multiplying it by 1.

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So let's start with the denominator.

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What is the denominator going to be equal to?

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The denominator is going to be equal to this squared.

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Once again, just a difference of squares.

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It's going to be 2 times the square root of y

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squared minus 5 squared.

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If you factor this, you would get 2 square roots of y plus 5

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times 2 square roots of y minus 5.

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This is a difference of squares.

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And then our numerator is 5y times 2 square roots of y.

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So it would be 10.

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And this is y to the first power, this is

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y to the half power.

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We could write y square roots of y.

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10y square roots of y.

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Or we could write this as y to the 3/2 power or 1 and 1/2

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power, however you want to view it.

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And then finally, 5y times 5 is plus 25y.

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And we can simplify this further.

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So what is our denominator going to be equal to?

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We're going to have 2 squared, which is 4.

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Square root of y squared is y.

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4y.

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And then minus 25.

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And our numerator over here is-- We could even write this.

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We could keep it exactly the way we've written it here.

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We could factor out a y.

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There's all sorts of things we could do it.

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But just to keep things simple, we could just leave

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that as 10.

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Let me just write it different.

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I could write that as this is y to the first, this is y to

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the 1/2 power.

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I could write that as even a y to the 3/2 if I want.

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I could write that as y to the 1 and 1/2 if I want.

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Or I could write that as 10y times the square root of y.

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All of those are equivalent.

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Plus 25y.

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Anyway, hopefully you found this rationalizing the

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denominator interesting.

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