Math Olympiad|Algebra simplification|No calculator is allowed

Hi, can we simplify this? Yes, let's go ahead and do it together. First, we can write this as The square root of the square root of nine is the same as three squared, right? minus The square root of eight is the same thing as four multiplied by two, okay. Now this is equal to the square root of now this we can see this we have three right minus this is the same thing as square root of four multiplied by square root of two, yes, because when you have the square root of a times b This is equal to the square root of a multiplied by the square root of b. Okay, so we continue. now this is equal to the square root of three minus the square root of two squared because four is the same as two squared multiplied by the square root of two, all right, so this is equal to The square root of three minus now this we can see this and we have two root two okay but three can be expressed as two plus one, right? So let's go ahead and write it here, so we can say that this is equal to the square root of two plus one minus two times the square root of two, okay? All right, now this is equal to the square root of, let's take this one to the other side, so we can write this as two minus two root two plus one, okay. All right, now from here, I want us to pay attention. Note that we can write as the square root of two raised to the power of two, right? Yes, and two root two is also equal to two multiplied by one multiplied by square root of two, and also one is the same thing as one squared Okay, all right, you're going to see what I'm doing soon. Now, when you have a minus b. raised to the power of two the algebra identity for this is equal to a squared minus two a b plus b squared, okay, yes, that is the algebraic identity for this. Now let's check what we have. under the radical sign, if we should replace this with everything we have here now, okay, so I'm I'm going to replace two with the square root of two squared, so I have the square root of two squared. Okay, minus two root two. Sentence two multiplied by one multiplied by root two, right? Minus two. multiplied by one multiplied by the square root of two plus now one is the same thing as one squared, right plus one squared, okay. Now, if we say, if we look at this and we say, um, root two minus one, if we should solve (√2 − 1)², okay? If we should expand it using this method in this case Our a is root two and our b is one, so it's going to be a squared, which is root two squared. Okay, minus two a b, which is two multiplied by a, which is root two multiplied by one, which is our b okay then plus one um one squared right plus one squared now if you look at this and look at What we have under the bracket under the root sign, you notice that we have the same thing. so that simply means that we can replace this with the square of root two minus one, so this is equal to the square root of that means I will write the entire thing as what square root of what the square root of two minus one squared, okay, now this is equal to, not that, this we cancel this, right Yes, and we have √2 minus 1. All right, so that simply means that if you simplify this square root of nine minus square root of eight is equal to square root of two minus One, thank you so much for watching. Please don't forget to give us a thumbs up if you actually enjoyed this video, please, it helps us to know that yes, you actually enjoyed our video, and if This is the first time you've seen this wonderful channel, so please click on the subscription. Click the button and turn on your notification bell so you don't miss our amazing videos Yes, so once we upload a video, we upload at least twice every day. They will notify you first. Thank you so much for watching.