Hello. Today we are going to look at simplifying thirds by adding or subtracting. All right, we've got five questions on the screen. So, let's get started. So, first thing we need to know is what are square numbers? What our square numbers are? Okay, so I would if I was you, list out all of the square numbers. So, that's 1 * 1 is 1 2 which is 1. 2 * 2 2 ^ 2 which is 4 and write them all the way out until 12. And the numbers in this yellow box are square numbers. Okay? Our goal with adding and subtracting thirds is to end up with square numbers underneath the square root sign because if we end up with square root with square numbers under the square root sign, we can simplify. So square<unk> of 4 is 2. Square<unk> of 9 is 3. See how it's a bit simpler to write them like that. But let's just start to practice. Now I've got our square numbers on the right hand side. That's what we're going to aim to get under the square root sign. Square root. So I'm just going to call it root 48. Oops. Root 48. Now you can write it as factors underneath the square root sign. And again, the goal is to get square numbers under the square root sign. I know that four is a factor of 48. So, I'm going to write that in there. 4 * 12 is 48. And then I can write this. I can actually write it further too because 4 is also a factor of 12. So, let's rewrite that. Rewrite 12 with a factor of 4. So four times four * 3. How about that? Okay. Then let's separate out the square root signs. So I'm going to write<unk> 4<unk>4 roo<unk>3 like that. And then I realize what is roo<unk> 4. Well, back here 2 * 2 is 4. Root 4 is 2. So that's what we're looking at. We can rewrite those. And there's invisible multiply signs in bene in between all of these thirds. Okay. So I can write two * 2 * <unk>3. And then our very last line. Let's multiply those twos. Two twos are four roo<unk>3. Now let's look at the next one. Root 20. Again we want square numbers underneath the square root sign. So this is roo<unk> 20 take roo<unk> 5 sorry. So let's write it. Square numbers under the square root. Well four is a factor of 20. That's nice. Oops. 4 * 5 and then the root five. That can't go any further. It's not a square number. So let's leave it like that. And then I can rewrite it separating into separate square root signs. and then roo<unk> 4 well four is a square number 2 * 2 is 4<unk> 5 stays the same now these roo<unk> fives these are going to be these terms are like terms right there's an invisible one here so what we're going to do we can just write 2<unk> 5 take 1 5 well 2 take 1 is just one one root 5 and we don't even need to write that one there. So that one is done. There is things called likeds which can help us simplify further which is nice. Next one 2<unk>25 take 3<unk> 45. Now bigger numbers seem scarier but it's okay. When 25 is in a number right 25 is listed there. Well 25 goes into 100 four times. So 25 goes into 125 five times. I'm only mentioning 25 because it is a square number. 5^ squar is 25. So that's why I'm thinking of breaking it down into factors. 2<unk> 5 * 25 take 3. Now what is happening to this 45? Let me have a think. Let me look at the square numbers less than 45 here. Are any of these are any of these numbers factors of 45? So, it helps to write these out every time as a guide. Are any of these numbers factors of 45? Well, one is, but that doesn't help us. Four, is that a factor? No. Nine is, though, so that's a nice little hidden factor there. 9 fives of 45. Now let's separate out those square root signs just to see everything. R<unk> 9<unk> 5. Now you're looking for the square numbers. See, I can see there's a square under a square root and a square under a square root there. That's what we want. So 2<unk> 5. nothing happens to roo<unk> 5 square<unk> of 25. 5 * 5 is 25. So let's just put that. And I have seen some students get a little bit confused. So just make sure you're writing that five without the square root sign. I've seen students write like<unk> 25 and then roo<unk> 5 * 5 and then they all get confused. It's like what what what's happening? But just stay calm. All right, take three. Now roo<unk> 9 is a square number. 3 * 3 is nine. So I'm going to write it without that square root sign. And then roo<unk> 5. Now that roo<unk> five is good because that means like thirds. So we like terms. So we can condense it further. But let's simplify first. Okay, I'm going to pull these out the front. these normal numbers because the order of multiplication doesn't matter. 2 * 5 is 10<unk> 5. Take 3 * 3 is 9<unk> 5. Have we made it as simple as we can? No, cuz there's like terms. So, let's do that. We're just going to put down roo<unk> 5 like terms. And then those numbers out the front. 10 take 9 is one. And we can write it without that one. We don't need it. Okay, powering through now. Slightly harder, but not really. And we might even Let's have a look. It's okay. Yeah, let's keep going. So, always looking for square numbers. Now straight up in this one here I can see the 36 is a square number. I can simplify that straight down. 66 is a 36. So write it without the square root sign. Root 36 is 6. Now let's have a look. 108. What are some factors? If we go through are there any factors here of 108? Now it's even, right? Um I don't know if you go through your times tables in your head, 9 12ves are 108. Um but yeah, there's a few different ways to do it. I guess I could see I could see four there, but four is also a factor, but um 9 is a higher factor. So you really want to be all over your times tables. Now roo<unk> 25 that is a square number five 5 to 25. So roo<unk> 25 is five and we can just write that straight down. Now you always want to be simplifying as you go. So we may as well add these normal numbers. 6 + 5 is 11. And now looking here root let's actually write we'll separate them out. 9<unk>2 and we've we've collected that five so I don't need to write it and now looking here 11 take well 9 is a square number<unk> 9 is 3 <unk>12 now that's an even number so that usually means you can write some factors out and get get down to square numbers okay so I know immediately that four one of these square numbers here four is a factor of 12. So I can do that and let's write it simpler. You don't have to separate out the square root signs every time. You can once you start getting faster at it, you don't have to do that. it. Uh, and then roo<unk> 4 is two. And I can see some like thirds in here. Whoops. 11. Take multiply those numbers out the front. -3 * So, take that sign out the front always. -3 * 2 is -6 or takeway 6. Now we're getting a lot simpler. Is it in simplest form yet? No. We've got some like thirds, so we may as well collect them. Here's our like thirds. So just the numbers out the front. So -6 take three. getting more negative on the number a number line<unk>3 and that's all you can do for that one. Now let's go to the next question the last question. All right now you're always looking again for square numbers under the square root sign. Even straight away I can see 49 is a square number 144 is a square number. Let's have a look. So 49 is a square number. 144. Let's try them out. 7 7s are 49. Nice. Plus now this 288 looks scary. But if you think about your square numbers, what are some factors? We can write this as 140 or as 2 * 144. Okay, looking for those looking for those square numbers. Now, roo<unk> 144 is 12. Looking down here, roo<unk> 18, it's an even number. Can we write it as a product of a square factor somewhere? Yes. 9 goes into it. So, you want to be all over your square numbers. Three sevens are 21. + 2 <unk>2<unk> 144 take 12 take 2<unk> 9 <unk>2 separating out the square root sign so we can see clearly and nothing happens to that roo<unk>2 roo<unk> 144 is 12 write it straight down and then roo<unk> 9. R<unk> 9 is three. And what else? Let's simplify. So simplifying between each step. Looking at this here, 2 <unk>2 * 12. We can grab those numbers out the front. 2 * 12 is 24 <unk>2 take 12 well I could have added that to the 21 but let's do it in the next step -2 * 3 is -6 <unk>2 now collect like terms normal numbers are like terms 21 take 12 is 9 and then We've got these are like terms. Take that sign out the front. So <unk>2 is a like term 24. So add the numbers out the front. 24 take six. 24 take six. I did not need my calculator for that, but that's all right. So it's 18 <unk>2. And that's all that you have to do. So I will stop the video