Everyone, earlier in our videos, we saw how to add and subtract like radicals. For example, if I take \(3\sqrt{5}\) and \(4\sqrt{5}\), I have the same number under the same index, so I can combine them. This \(3\) and \(4\) just becomes \(7\). You just add the little numbers in front, and I have \(7\sqrt{5}\), and that is the answer. What I'm going to show you in this video is that sometimes you'll have radicals and square roots that aren't similar, like \(\sqrt{5}\) and \(\sqrt{20}\), and you're going to have to add them. I'm actually going to show you how we do that in this video, and we're going to use some old ideas that we've already seen before. Let's go ahead and get started. The idea is that when you're adding/subtracting radicals that aren't alike, you're going to have to simplify them first. So, simplify before you end up combining them like radicals. Here's the idea. Can I simplify \(\sqrt{5}\)? In other words, can I pull out a perfect power? Well, no, because the only factors of \(5\) are \(1\) and \(5\). Can we take the \(20\) and break that down into a perfect power? Well, actually, yes, we can because we saw how to do this. I'm going to take the \(20\), and I'm going to break this down into a product. And the hope is that one of the radicals ends up being a perfect power. So, we looked at \(4\), \(9\), and \(16\) all the way up to \(20\), and we saw that \(16\) and \(9\) didn't work because they don't go into \(20\), but \(4\) does. So, in other words, I can take this radical \(20\) and break it up into the product of \(\sqrt{4}\) and \(\sqrt{5}\). So why is this helpful? Well, because this just becomes \(\sqrt{5} + 2\), and this whole thing just becomes \(2\sqrt{5}\). So in other words, I've ended up or started with two square roots that were unlike each other. But by simplifying down, now I've turned it into a problem where I have the same radicands in the same index. In other words, I started out with unlike radicals. And if you simplify it, you'll end up with an expression that is like radicals. And this basically just turns into the problem on the left. So, how do I add these things? Well, \(\sqrt{5} + 2\sqrt{5}\). You just add the numbers in front. This is just \(3\sqrt{5}\), and that's how to solve these kinds of problems. Alright? So, break them down before you start combining them. That's really all there is to it. Let's go ahead and take a look at a couple more examples here. Here, we're going to do \(5\sqrt{2}\) and \(\sqrt{18}\). Alright? So what happens is, can I simplify \(\sqrt{2}\)? Well, no. Well, the whole idea is that I want these two square roots to eventually be the same because then I can subtract them. So I can't simplify \(\sqrt{2}\), but can I simplify the \(\sqrt{18}\)? And we saw how we can do that by pulling out a perfect square. In this case, \(4\) isn't going to work, but \(9\) will. So, I'm going to break this up into a product of two radicals. \(9\) goes into \(18\), and I get \(2\) as what's left over. So again, what we end up with here is we end up with \(5\sqrt{2} - 3\). That's what this becomes, \(3\sqrt{2}\). So again, they were unlike radicals first. Now I've simplified them, and they turn into like radicals. So now I can just go ahead and subtract them. \(5 - 3\) just becomes \(2\sqrt{2}\), and that's the answer. Alright, so what I want to do if you have a calculator handy is actually want you to plug this expression into your calculators. \(5\sqrt{2} - \sqrt{18}\). You're just going to get a number, and if you do \(2 \times \sqrt{2}\), you're going to get the exact same number. Alright. So, this is just another way, a simpler way to write that expression that we started with. Alright. That's the whole idea. Let's take a look at this last one over here. \(\sqrt{18}\) and \(\sqrt{50}\). Alright. Same idea. I can only add them when these two square roots are like each other. So here's the question. Can I take the square roots, and can I simplify it so that I get a \(\sqrt{50}\) out of it? Well, no, because if we break it down, all the numbers get smaller. But can I take the \(\sqrt{50}\) and break it down so that I get a \(\sqrt{18}\) out of it? Because then I would be able to add them. Well, let's try that. Let's try to break down this \(\sqrt{50}\) into a product of \(2\) terms, which you get an \(18\). So, basically, what you're asking is, is \(50\) divisible by \(18\)? Well, if you do \(18 \times 2\), that's \(36\). And if you do \(18 \times 3\), that's \(54\). So \(18\) doesn't evenly go into \(50\). So, in other words, I can't sort of break this thing up into a product where I get \(18\). So how do I solve this problem? Well, it turns out that unlike these sorts of problems over here, where we only had to break down one of the terms, in problems, you might actually have to simplify both of the terms before you can start combining them. So we're going to have to break down the \(18\) and the \(50\), and the hope is that you're going to get two radicals that are the same. Alright? So that's the idea. So we've actually already seen how to break up the \(\sqrt{18}\) in the other problem in part A. We saw that this just breaks down into \(\sqrt{9} \times \sqrt{2}\). What happens to the \(50\)? Well, if you go down the list, \(4\) doesn't go into \(50\), \(9\) doesn't either. \(16\) doesn't go into \(50\). What about \(25\)? \(25\) does go into \(50\). So in other words, it just becomes \(25 \times \sqrt{2}\). So now what you've seen here is by breaking down the \(\sqrt{18}\) and the \(\sqrt{50}\), when you factor it out, and you pull out perfect powers, basically you end up with the same radicals. Alright. So let's clean this up a little bit more. What does \(9\sqrt{2}\) become? That just becomes \(3\sqrt{2}\). And then what does the \(25\sqrt{2}\) become? That just becomes \(5\sqrt{2}\). Remember, we had an addition sign over here. So now I've basically ended up with two radicals that are alike, So now I can add them, and this whole thing just becomes \(8\sqrt{2}\). Alright. So again, if you have a calculator handy, go ahead and plug in this expression in your calculator. \(\sqrt{18} + \sqrt{50}\). You're going to get a number. I think it's something like, I think it's like \(11.7\) or something like that. And if you plug in \(8\sqrt{2}\), you're going to get that exact same number. Alright? So that's it for this, folks. Thanks for watching. I'll see you in the next one.