 In this video, I'm going to be talking about adding and subtracting radicals. So we're adding and subtracting square roots. Got a couple of examples up here. I'm going to go through multiple number examples. I'm not going to use any algebra here. I'm not going to use any variables. I'm just going to use numbers for these examples. Okay. Adding and subtracting radicals is actually relatively simple. It's actually very similar to adding and subtracting your variable expressions. What am I talking about? Variable expressions. For example, to do something else here, if I had 3x plus 5x. Okay. And you just want to simplify that. What you're going to do, since I have an x here and an x there, they are like terms. And so what I'm going to do is I'm going to add the coefficients. Three and five, the vocab here, are the coefficients in front of x. So what I'm going to do is I'm going to simply add those together. Three plus five is 8x. You can think of it as 3x's and 5x's make a total of 8x's. Okay. So we're going to take that same concept and we're going to apply it to radicals. All right. So basically the only rule that you really have to worry about here is you need to, if they have the same radical, then you need to add the coefficients together. So for example, three root five plus 10 root five. So I look at the roots. I have a root five here and a root five here. That means that these two terms are alike. That means I can add them together. So I'm going to take the coefficients, the numbers out front, and I'm going to make this a 13 root five, a little bad three there. This is going to be 13 root five. So if I have the same radicals, I simply need to just add or subtract the coefficients together. And I say subtract because if I look to my right, I have this kind of the same terms, but instead of adding them together now, I'm subtracting them. Okay. But I still go through the same process. Do I have the same radicals? Yes. That means I take the coefficients and add or subtract them. So three minus 10 is a negative seven root five. Negative seven root five. And yes, it's okay to have negative outside of a radical. All right. Those are two examples. And again, just to kind of summarize, if I want to add or subtract radicals, I need to add or subtract them like I do regular variables. I just look to see if they have the same radical, and then I add the coefficients together. But what if we don't have the same radical? So I'm going to do a couple more examples if I don't have the same radicals. So for example, what if I had the square root of 80 minus five root five? Okay. Some students would take a look at this and throw up their hands and say, oh, can't do it. I see a square root of 80 and I see a square root of five. Those are not the same. Can't do it. Well, what you have to keep in mind is that larger numbers underneath the radical, we can actually simplify those. So when you look at that square root of 80, I can actually simplify that a little bit. So what I can do is, going back to simplifying our radicals, I need to find the largest perfect square that divides evenly into 80. And actually, 80, I can split this up into the square root of 16 and the square root of five. 16 goes into 85 times. 16 times five gets me back to 80. Now, the reason we want to use 16 is because it's the largest perfect square that divides evenly into 80. So I always want to use the largest perfect square. Now, the reason we want to do that is because square root of 16 is four. Square root of five, you don't know what that is, so it just stays square root of five. So now, instead of 80, I have a four root five. So what I'm going to do is bring this down also. So four root five minus five, root five. And now, we have the same radicals. So now what I'm going to do is, again, go through the same process. Do I have the same radicals? Yes, I have a root five and I have a root five. Now, what I need to do is I need to add or subtract the coefficients. In this case, subtract four minus five, which is a negative one. Now, notice the radical doesn't change. It just stays square root five. But the coefficient out front becomes negative one. Now, you can either leave it as negative one root five or you can have it as negative root five. Both of those answers are the same because negative one times root five and then a negative root five will actually give you the same answer. So either one of those answers is OK. For those of you just learning this, this answer here on the left is going to be your best bet because you can actually visualize, you can actually see that negative one in progress. OK, one more example, one more quick example, five root 32 plus 15 root two. OK, so here's another example. Now, again, you got to think to yourself, if I want to combine these radicals, if I want to add them together, I need to have the same radical. Well, the first one is a radical 32. The second one is a radical two. So those aren't the same. So I need to try and see if I can simplify one of them. The square root of two over here is already simplified down as far as we can go. But the square root of 32, I can actually continue to simplify that down. So I'm going to take 32, split it up and simplify it down. Now, what causes a little bit of trouble is this five here, but I just have to just bring the five down with all the simplifying that I do, and it won't cause much trouble at all. So this is going to be five times the square root of 16 times the square root of two. 16 times two gives me back to 32. So 32 gets split up into 16 and two. Now, the reason I chose 16 is because we actually know what the square root of 16 is, it is four. So this is five times four times the square root of two. I don't know what the square root of two is, so again, I just leave it. All right, and then, just look like I can simplify this any more other than taking four times five, which gives me 20 root two. Bring the rest of this down. Now we notice that root two, root two, those are the same roots. So I need to add the coefficients out front. This gives me 35 root two. And there's my simplified answer. So again, when you're adding or subtracting radicals, make sure that you have the same radicals. And if you have the same radicals, then you simply just add or subtract the coefficients, the numbers out front. And then if you see a problem like this, where you don't have the same radicals, make sure that you simplify down the larger ones. So in this case, I have a radical two and a radical 32. That radical 32, I simplified it down and eventually I got a radical two and I was able to add these together. So make sure when you look at problems like this, you simplify it down as far as you can before you decide whether or not you can combine your terms, whether or not you can add or subtract your terms. So again, to recap, to summarize, when you're adding or subtracting radicals, make sure you have the same radical and you simply just add or subtract the coefficients.