Welcome back, everyone. We saw how to rationalize the denominator by taking something like 1 over radical 3. We multiply the top and the bottom by whatever was on the bottom, and we end up undoing the radical, and that was awesome. We could do that. So, what I'm going to show you in this video is that sometimes that won't happen. You might have a problem that looks like 1 over 2 + radical 3, in which you actually have 2 terms in the denominator, and when the denominator has 2 terms, multiplying by the same radical won't actually eliminate it. So what I'm going to show you is, actually, we're going to need something else to rationalize the denominator. We're going to need something called the conjugates. Rather than tell you, let me just go ahead and show you. Let's just jump into our problem here.
So why can't we just use 2 +3 on the bottom? Well, if you end up doing 2 +3, then when you multiply across, this actually ends up being a binomial multiplied by a binomial. So in other words, we actually have to foil. And if you foil this out, what's going to happen is, on the bottom, you're going to get 2 times 2. That's the first. We're going to get 2 times radical 3. Those are the outer terms. Same thing for the inner terms. So, in other words, you end up with 4 radical 3. And on the inner terms, radical 3 times itself will just be 3. So, in other words, I multiplied it by itself on the top and the bottom, but I still ended up with a radical on the bottom. I didn't get rid of that, and, remember, that's bad. You can't have radicals on the bottom. So multiplying it by itself is not going to work here. So what do we do? Well, instead of multiplying by itself, we do something. We multiply by what's called the conjugate of the bottom. Basically, what the conjugate is is you're just going to reverse the sign between the two terms. So, for example, if I have 2 +3, then the conjugate is just going to be 2 minus radical 3. That's the conjugate. You just take the sign between the terms and you flip it. So the general formula is if you have something like a + radical b, then the conjugate is going to be a minus radical b and vice versa. Those two things are conjugates of each other.
So what do we do here? Well, I'm just going to rewrite this. 1 over 2 +3. Now we're going to multiply not by itself. We're going to multiply by the conjugate, so in other words, 2 minus radical 3. And remember, whatever we multiply on the bottom, we have to multiply on the top. It has to be the same thing. So why does this work? Well, if you notice here, we're actually multiplying the same exact terms, 2 and radical 3, except the signs are flipped between them, and this actually ends up being a difference of squares. So remember how a difference of squares works? Basically, what happens is that you square these two numbers, so 22 becomes 4, and you square these two numbers, and then you basically just stick a minus sign between them. So, in other words, radical 3, when you square, it just becomes 3, and this is just a difference of squares. And on the top, when you multiply straight across, this ends up being 2 minus radical 3. Alright? So what do you end up with? You end up with 2 minus radical 3 divided by just 1. Now you won't always get 1 here. We just got 1 because we had 4 - 3. But, basically, what happens is that we just got rid of the radical. So, in other words, we've rationalized the denominator here. So multiplying by multiplying a radical by its conjugate, in fact, always eliminates the radical. That's why it's super useful, and it always just results in a rational number, like 1 or something like that. So just let me summarize really quickly here. When you have a one-term denominator, you multiply the top and the bottom by whatever is on the bottom. And when you have a 2-term denominator, you multiply by the conjugate of the bottom. Alright? But these are just the 2 ways that you rationalize the denominator. Hopefully, that made sense. Let me know if you have any questions. Thanks for watching.