 Hi, this is Chicho, back again. There's a couple problems that I wanted to take care of, just finish off the Explanations of Radicals section that appear a lot in a lot of exams that I've seen from different types of schools and stuff like this. And basically it's a type of problems that they really like giving because they look difficult but they're extremely simple and it's just basically radicals within radicals. For example, if you had square root of 2 plus the square root of 4, that's a typical type of problem that they give. These are really simple but for some reason, having a square root symbol inside a square root symbol really throws people off. Now, the way this works is I'm assuming you guys, or if you're dealing with this kind of stuff, you've already gone through your bed mass, the principle, where you're talking about brackets, exponents go first and then multiplication, division, addition, subtraction. So I'm assuming you know the order of operations. I'm assuming you know the order of operations. So the way this would work is if you get something like this where there's addition and subtraction inside a radical, you have to deal with the inside first. So what you would do here is 4 is 2, 2. 2 has come out as a 2. So this would be square root of 2 plus 2. 2 plus 2 is 4. So this becomes the square root of 4. Square root of 4 is just 2. So the answer is 2. Really straightforward. Let's do a little bit more complicated one. So let's say they gave you something larger. Let's say they gave you a plus. What have we got? Square root of 4 plus. Square root of 16 plus 2 squared. So what we've got here is multiple root symbols within root symbols. I've got a radical there and radicals are just exponents. So the way you do this is you deal with each one separately and then when you combine your like terms and at the end you take the square root of the whole thing. So first thing you do is square root of 4. Let's rewrite this here. So the 8 we can't do anything yet. So we just write down the 8 as an 8. We've got plus. Square root of 4, we should already know that. That's just 2. Over here you've got the square root of 16. Square root of 16, if you break it down it goes 4 times 4 and you don't have to break this down to its prime numbers because you already found a pair. Square root means you're looking for pairs. Clones. If this was a cube root you'd be looking for triplets. But you already got a pair so you don't have to break it down anymore. So as soon as you get a pair with the square root circle it, you're done expanding that tree. So square root of 16 is 4 and square root of 4 is just 2. It's just the same thing here. So that becomes plus 2. 2 squared is 4. So now what you've got is square root of 8 plus 2 is 10 plus 2 is 12 plus 4 is 16 square root of 16 and square root of 16, we already did it over there is just 4. So this whole thing turns into 4. Now keep in mind you don't just have to have square root symbols, right? You could have cube root, the fourth root or something like that. Let's just do one more where it's just not a square root. It's a cube root or something like that, okay? So I just made up another larger problem with the cube root the fourth root or something like that. So let's say you had something like this. You had the cube root of the fourth root of 16 plus. What are we going to do? The square root, cube root of 8 squared plus the fifth root of 32 all squared. Okay? Well hopefully you can see that. So this is the cube root of the fourth root of 16 plus blah blah blah blah blah. You've got the whole thing here, right? Now these guys are separated by additions, right? Each one you're going to have to deal with separately because you can't combine anything right now. Everything's all to the different roots and different powers. So what you've got to do is do each one separately and then combine at the end and see if you can take the cube root out. So what we've got here is you write down your cube root symbol here. The fourth root of 16, this breaks down to 4 times 4. 4 is 2, 2, 2, 2. So you're looking for the fourth root. You've got 4 twos right there, right? So the fourth root of 16 is just 2 plus. You've got the cube root of 8. 8 is just 2, 2, 2. So that's the cube root. So that's going to be cube root of 8 is going to be 2. 2 squared is going to be 4. And the square root of 4 is just going to be 2 plus. You've got the fifth root of 32. The fifth root of 32, this breaks down to 4 times 8. 4 is just 2, 2. 8 is 2, 2, 2. So fifth root of 32 is just 2. 2 squared is going to be 4. So that's going to be 4. So what you've got now is the cube root of... The cube root of... I shouldn't go so far anyway. Because that's just going to break down to the last term, right? So it's going to be 2 plus 2 is 4, plus 2 is 8. So it's going to be the cube root of 8, which is... Well, we should know by now what the cube root of 8 is. So cube root of 8 breaks down to 2 times 2 times 2. So cube root of 8 is just going to be 2 again. So these are just a set of, just a branch of questions that most teachers like to give in exams. Because they test the basic operations, the basic radicals and the basic exponents where you're taking the squaring it, cubing it, or whatever you're doing it. Yeah, so deal with these, they're not very difficult. They look ugly to begin with. Actually, they don't look that ugly. I actually like the way they look. So they just, you know, they tend to throw people off because people want to just, you know, go from the problem, go from the question all the way to the answer in one step. And you can't do that. You have to crunch things down until you can get them into like terms and then you can deal with it, okay? Again, this is just simple radicals. You should be able to do it in grade 9. Yeah, definitely in grade 9. Most definitely in grade 10. And this is the type of stuff that you end up getting in grade 11 and 12, you know, when you start dealing with logs and stuff like that, okay? So that's just one branch of questions you get when it comes to radicals.