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- In this video, we're going to learn how to rationalize the
- denominator.
- What we mean by that is, let's say we have a fraction that
- has a non-rational denominator, the simplest one
- I can think of is 1 over the square root of 2.
- So to rationalize this denominator, we're going to
- just re-represent this number in some way that does not have
- an irrational number in the denominator.
- Now the first question you might ask is,
- Sal, why do we care?
- Why must we rationalize denominators?
- And you don't have to rationalize them.
- But I think the reason why this is in many algebra
- classes and why many teachers want you to, is it gets the
- numbers into a common format.
- And I also think, I've been told that back in the day
- before we had calculators that some forms of computation,
- people found it easier to have a rational number in the
- denominator.
- I don't know if that's true or not.
- And then, the other reason is just for aesthetics.
- Some people say, I don't like saying what 1
- square root of 2 is.
- I don't even know.
- You know, I want to how big the pie is.
- I want a denominator to be a rational number.
- So with that said, let's learn how to rationalize it.
- So the simple way, if you just have a simple irrational
- number in the denominator just like that, you can just
- multiply the numerator and the denominator by that irrational
- number over that irrational number.
- Now this is clearly just 1.
- Anything over anything or anything over that same thing
- is going to be 1.
- So we're not fundamentally changing the number.
- We're just changing how we represent it.
- So what's this going to be equal to?
- The numerator is going to be 1 times the square root of 2,
- which is the square root of 2.
- The denominator is going to be the square root of 2 times the
- square root of 2.
- Well the square root of 2 times the square
- root of 2 is 2.
- That is 2.
- By definition, this squared must be equal to 2.
- And we are squaring it.
- We're multiplying it by itself.
- So that is equal to 2.
- We have rationalized the denominator.
- We haven't gotten rid of the radical sign, but we've
- brought it to the numerator.
- And now in the denominator we have a rational number.
- And you could say, hey, now I have square root of 2 halves.
- It's easier to say even, so maybe that's another
- justification for rationalizing this
- denominator.
- Let's do a couple more examples.
- Let's say I had 7 over the square root of 15.
- So the first thing I'd want to do is just simplify this
- radical right here.
- Let's see.
- Square root of 15.
- 15 is 3 times 5.
- Neither of those are perfect squares.
- So actually, this is about as simple as I'm going to get.
- So just like we did here, let's multiply this times the
- square root of 15 over the square root of 15.
- And so this is going to be equal to 7 times the square
- root of 15.
- Just multiply the numerators.
- Over square root of 15 times the square root of 15.
- That's 15.
- So once again, we have rationalized the denominator.
- This is now a rational number.
- We essentially got the radical up on the top or we got the
- irrational number up on the numerator.
- We haven't changed the number, we just changed how we are
- representing it.
- Now, let's take it up one more level.
- What happens if we have something like 12 over 2 minus
- the square root of 5?
- So in this situation, we actually have a binomial in
- the denominator.
- And this binomial contains an irrational number.
- I can't do the trick here.
- If I multiplied this by square root of 5 over square root of
- 5, I'm still going to have an irrational denominator.
- Let me just show you.
- Just to show you it won't work.
- If I multiplied this square root of 5 over square root of
- 5, the numerator is going to be 12 times the
- square root of 5.
- The denominator, we have to distribute this.
- It's going to be 2 times the square root of 5 minus the
- square root of 5 times the square root of 5, which is 5.
- So you see, in this situation, it didn't help us.
- Because the square root of 5, although this part became
- rational,it became a 5, this part became irrational.
- 2 times the square root of 5.
- So this is not what you want to do where you have a
- binomial that contains an irrational number in the
- denominator.
- What you do here is use our skills when it comes to
- difference of squares.
- So let's just take a little side here.
- We learned a long time ago-- well, not that long ago.
- If you had 2 minus the square root of 5 and you multiply
- that by 2 plus the square root of 5, what will this get you?
- Now you might remember.
- And if you don't recognize this immediately, this is the
- exact same pattern as a minus b times a plus b.
- Which we've seen several videos ago is a
- squared minus b squared.
- Little bit of review.
- This is a times a, which is a squared. a
- times b, which is ab.
- Minus b times a, which is minus ab.
- And then, negative b times a positive
- b, negative b squared.
- These cancel out and you're just left with a
- squared minus b squared.
- So 2 minus the square root of 5 times 2 plus the square root
- of 5 is going to be equal to 2 squared, which is 4.
- Let me write it that way.
- It's going to be equal to 2 squared minus the square root
- of 5 squared, which is just 5.
- So this would just be equal to 4 minus 5 or negative 1.
- If you take advantage of the difference of squares of
- binomials, or the factoring difference of squares, however
- you want to view it, then you can rationalize this
- denominator.
- So let's do that.
- Let me rewrite the problem.
- 12 over 2 minus the square root of 5.
- In this situation, I just multiply the numerator and the
- denominator by 2 plus the square root of 5 over 2 plus
- the square root of 5.
- Once again, I'm just multiplying the number by 1.
- So I'm not changing the fundamental number.
- I'm just changing how we represent it.
- So the numerator is going to become 12
- times 2, which is 24.
- Plus 12 times the square root of 5.
- Once again, this is like a factored
- difference of squares.
- This is going to be equal to 2 squared, which is going to be
- exactly equal to that.
- Which is 4 minus 1, or we could just-- sorry.
- 4 minus 5.
- It's 2 squared minus square root of 5 squared.
- So it's 4 minus 5.
- Or we could just write that as minus 1, or negative 1.
- Or we could put a 1 there and put a
- negative sign out in front.
- And then, no point in even putting a 1 in the
- denominator.
- We could just say that this is equal to negative 24 minus 12
- square roots of 5.
- So this case, it kind of did simplify it as well.
- It wasn't just for the sake of rationalizing it.
- It actually made it look a little bit better.
- And you know, I don't if I mentioned in the beginning,
- this is good because it's not obvious.
- If you and I are both trying to build a rocket and you get
- this as your answer and I get this as my answer, this isn't
- obvious, at least to me just by looking at it, that they're
- the same number.
- But if we agree to always rationalize our denominators,
- we're like, oh great.
- We got the same number.
- Now we're ready to send our rocket to Mars.
- Let's do one more of this, one more of these right here.
- Let's do one with variables in it.
- So let's say we have 5y over 2 times the square
- root of y minus 5.
- So we're going to do this exact same process.
- We have a binomial with an irrational denominator.
- It might be a rational.
- We don't know what y is.
- But y can take on any value, so at points it's going to be
- irrational.
- So we really just don't want a radical in the denominator.
- So what is this going to be equal to?
- Well, let's just multiply the numerator and the denominator
- by 2 square roots of y plus 5 over 2 square
- roots of y plus 5.
- This is just 1.
- We are not changing the number, we're just
- multiplying it by 1.
- So let's start with the denominator.
- What is the denominator going to be equal to?
- The denominator is going to be equal to this squared.
- Once again, just a difference of squares.
- It's going to be 2 times the square root of y
- squared minus 5 squared.
- If you factor this, you would get 2 square roots of y plus 5
- times 2 square roots of y minus 5.
- This is a difference of squares.
- And then our numerator is 5y times 2 square roots of y.
- So it would be 10.
- And this is y to the first power, this is
- y to the half power.
- We could write y square roots of y.
- 10y square roots of y.
- Or we could write this as y to the 3/2 power or 1 and 1/2
- power, however you want to view it.
- And then finally, 5y times 5 is plus 25y.
- And we can simplify this further.
- So what is our denominator going to be equal to?
- We're going to have 2 squared, which is 4.
- Square root of y squared is y.
- 4y.
- And then minus 25.
- And our numerator over here is-- We could even write this.
- We could keep it exactly the way we've written it here.
- We could factor out a y.
- There's all sorts of things we could do it.
- But just to keep things simple, we could just leave
- that as 10.
- Let me just write it different.
- I could write that as this is y to the first, this is y to
- the 1/2 power.
- I could write that as even a y to the 3/2 if I want.
- I could write that as y to the 1 and 1/2 if I want.
- Or I could write that as 10y times the square root of y.
- All of those are equivalent.
- Plus 25y.
- Anyway, hopefully you found this rationalizing the
- denominator interesting.