 This video is about rationalizing the denominator. This is a process that we use to simplify radicals. Sometimes it can be a little bit of a difficult process just because you have to remember one step to do, and then you've got to combine a lot of terms, a lot of numbers together. But for the most part, it's a pretty straightforward process. So why do we have this process? Well, first of all, if we look at this number, the square root of 3 times 5 divided by the square root of 2, then this right here is what causes us trouble. Dividing by the square root of 2. Now why is that trouble? Think of it this way. The square root of 2 is an irrational number. Irrational numbers are basically decimals that keep repeating and go on forever. And so if we divide by an infinitely repeating decimal, that causes trouble. You can't really divide by an infinite set of numbers. That just doesn't make any sense. So what we do is we're going to take this square root of 2 and we're going to turn it from an irrational number into a rational number. That's why we call it rationalizing the denominator. Remember your vocabulary, denominator is the bottom of the fraction. Okay, so how do we do that? Well, we have to look at what is causing us trouble. The square root of 2 is the number that's causing us trouble. So we are going to multiply by what's causing us trouble. We're going to multiply on the top and on the bottom by the square root of 2. Where do we get this? We get this from the bottom of our fraction. Now another question I get is why do we do this? Why does this work? Now technically what we are doing is we're simply just multiplying times 1. We are taking the square root of 2 divided by the square root of 2. This red right here is simply just 1. So all we're doing is we're taking this fraction times 1 but what's going to end up in the end is we're going to rationalize this denominator. We're going to turn this from an irrational number into a rational number. We're going to change it up a little bit so it's a better looking number on the bottom. Now at the end of this video I'm going to go over why this process works, why we can't do this, but I'm going to wait until the end of the video to do that. But technically we're just multiplying times 1. Moving on, what we're going to do now is we're just going to multiply this. I'm going to bring everything together. On top I have 3 times the square root of 5 times the square root of 2 divided by the square root of 2 times the square root of 2. Now what I want to do is I want to combine anything that I can. Now this 3 and the root of 5, you cannot combine those. You can't multiply them. One is outside of the radical. One is inside the radical. Three is outside the radical. Five is inside the radical. You can't cross that boundary and multiply those two numbers. It just doesn't work. But on the other hand, if I have the square root of 5 and the square root of 2, both of those are underneath the radical. Which means I can actually multiply those numbers since they are both underneath the radical. Refer back to the product property of square roots is why we can do this. So what I can do is I can multiply these together. 5 and 2 make 10. So I have 3, root, 10. And yes, when you multiply it, it stays underneath the radical. So that's what the top looks like. The bottom, 2 times 2, or I should say the square root of 2 times the square root of 2, is the square root of 4. Now when you look at this, that should pop in your memory. Oh, I know what the square root of 4 is. The square root of 4 is actually, simplify this down a little bit more, 3, root, 10 over 2. And that's it. That's as far as we can go. There's no more simplifying that we can do. 2 does not divide evenly into 3, so we just leave it alone. Now 2 divides evenly into 10, but 2 does not divide evenly into the square root of 10. So again, I can't cross that boundary. Just like over here. 3 times the square root of 5, I can't cross that boundary. Same thing here. 2 and root 10, I can't cross that boundary. They're not the same. 1's a number, 1's a radical. So I can't divide these numbers. So that right there, that's as good as it gets. 3 times the square root of 10 divided by 2. So notice what we've done here. We have taken this irrational number of the square root of 2, and we have turned it into a rational number of 2. This is a much nicer, much neater fraction to work with. So we've taken an irrational number, and we've rationalized the denominator turning it into 2. So that's rationalizing the denominator, that's the process. Now what I'm going to do is I'm going to go into a little bit extra of why we do this, why this works. Technically what we're doing is we're multiplying by 1, so this isn't actually very difficult to understand. I'm just going to use simple numbers to help me with this. So if I take 5 times 1, I'm still just going to get 5. Not a big deal, everybody knows that. 5 times 1 is equal to 5. But what I can also do is instead of just multiplying times 1, what if I multiply something that's equivalent to 1? What if I multiply it by something that is the same as 1? So let's use 5 times 3 over 3. What's that going to be equal to? 3 over 3, that's the same thing as 1. We really haven't changed much of anything. So I'm going to multiply these together. 5 times 3 is 15. And there's a 1 on the bottom here. 1 times 3 is 3. 15 divided by 3 is what this is. 15 divided by 3, so this is simply 5. So notice we've gone from 5, we multiply times 3 over 3 and we got 5 out of it. So again, nothing has really changed. The value of the number has not changed. When we multiply times 1, so that's why rationalizing the denominator, that's why this process works, is because all we're doing is we're just multiplying times 1. We're just multiplying times 1. So the value of the number does not change. So from beginning to end, the value of the number is not changing. It's only the look of the number that's changing. So this will be similar to this bottom. So we're taking 5 times 3 over 3 and we get 15 out of 3. So it looks different, but it is still 5 in the end. So that's why this process works. Also one thing to note is if you look at the bottom, you're always going to get a whole number when you multiply this way. I'll show more examples in a different video, but you're always going to get a whole number when you multiply by this. And this process is going to work every time, as long as you always use this thing on the bottom, this number on the bottom that is causing us trouble. In this case, the square root of 2 is causing us trouble, so we multiply by the square root of 2 over 2.