A math olympiad algebra simplification| can you simplify this?

Hi, let's solve for the value or let's simplify what we have on the board. Okay, let's first of all focus on what we have under the radical sign now this is equal to the square root of One is the same as two divided by two, right, because I want to have the same denominator. so two divided by two plus what root three divided by two that means that this is equal to the square root of now we have the same denominator so write that and two plus what root three two plus root three, all right? Now I want to introduce something because I have a target. Okay, they say simplify, so I should do everything possible to simplify this. Okay, so I want to because two divided by two will still give me one, so I haven't really introduced anything so I can write this to be equal to the square root of; sorry, trying to get a straight line okay now the square root of what two divided by two multiplied by what two plus root three divided by two, all right. Now that we have this, let me turn this into multiplication now since we have this, this is equal to the square root of two times this two, we give us four plus two times Root three, we give us what two root three divided by two times two, we still give us four, okay all. Right now, I want us to analyze this for those who might be confused—this two over two multiplied. two plus the square root of three over two, okay, is the same thing as the numerator, right? We multiply the numerator as well, which is two plus root three, okay? So if I expand this bracket, two times three Give me four and two times three, three. Give me two root three. Then these two, we multiply these. Two to give us four, okay? So I just want to flare the air. All right, now I want us to concentrate. four plus two root three, let's simplify it first, so four plus two root three, okay now This can be written as three plus one plus two root three, okay, so three plus one plus two. root three and this is equal to three plus two root three plus one, all right? Yes, I know where. I'm driving at, if this is the first time you're coming across this channel, please click on the Subscribe button on your notification bell so I'll be notified anytime I upload a new video Okay, so I don't miss any of our amazing videos, and please give us a thumbs up if you're actually Enjoying this video? Okay, thank you so much. All right, now what am I trying to form? Recall. Recall that a plus b squared is equal to. Sorry, let me turn to this side. now a plus b squared is the same thing as a squared plus two a b plus b squared, all right, is the same thing. Now I want us to write what we have here. This is what we have: we have three plus two root three plus one. Now I can write three as what? the square root of three squared, right? Yes, I can write the square root of three squared, right? Because This will divide this, and we are left with just three, which is this plus two root three. plus this one is the same thing as what one squared, right? Yes, all right. Now, if we look at this and We say the square root of three is equal to a, and that one is equal to b. You can say that if we say. (root three plus one) squared using this particular identity that this will be equal to root three squared root three squared plus two a b right that is two multiplied by r a which is root three multiplied by r b, which r b is one, then plus b squared, which is one squared Now, if you solve this, you will have root three squared. We give you what this is; you can see this. You have three right plus two root three right two root three plus one squared will give us one Yes, now when you look at this, you see that you've gotten this back. Now, what am I trying to what I'm trying to say is that this is actually equal to root three plus one squared, so another A way of writing this is to write root three plus one squared. Therefore, I'll come back to our equation right and I said record that all this while I've been solving four plus two root three That's what we've been analyzing. All right, this is four plus two root three, so I can replace this. four plus two root three since this is all we've analyzed I can replace it with what root three plus one squared, all right, without causing any confusion, so this implies that, permit me to clean this part of the board, so this implies that the square root of four plus two times the square root of three divided by four is equal to the square root of, so I'm going to replace this now with this root three plus one all squared divided by what four now I can write this as what the square root three plus one all squared divided by two squared because four is the same thing as two squared Okay, now this is equal to the square root of the quantity root three plus one, divided by two. Now, before I write anything, recall that a squared or a to the power of b—where should I place the exponent? Yeah, it will enter here a to the power of b divided by, say, c to the power of b, okay. Is the same thing as a divided by c raised to the power of b once they have the same power, right? so that simply means that we can rewrite this as what this is root three plus one over two Everything squared since they have the same power. All right, now permit me to clean this side of the board, we hope you've given us a thumbs up; please do so if you haven't already, so that this is equal to what the square roots of, so now we have this; we divide this, or this—you can see this. Your square roots always cancel your square, so what we have left is the square root of three. Plus, sorry, we've already canceled the square roots, right? Yeah, we have canceled the square roots. so what we have left is the square root of three plus one divided by two, so this implies that The square root of one plus the square root of three divided by two is the same as the square root of three. plus one divided by two, so this is the simplified path or the simplified version Thank you so much for watching, have a lovely day.