Algebra simplification| Maths olympiad.

Hi, lovely people, we're asked to simplify this without a calculator now. This is all I promise you. By the end of this video, whenever you see something like this, You should be able to solve it or at least attempt it to some extent. Yes, because I'm going to give you the easy step-by-step method. You are going to use it to solve this or to simplify this, rather. OK, now the first thing you need to do is whatever you see inside that bracket, Call it X, call it Y, call it any letter you want to simplify it first. Alright, so let X be equal to 1 plus the square root of 5, divided by 2. I'm going to explain what I'm doing after a few steps. For now, just trust the steps. OK, now that you have done this, you need to cross-multiply. Why are you doing that? Because you actually want to remove this root from here. So let's cross multiply. If we cross-multiply, we get 2X equals 1 plus the square root of 5. Now we are going to move this constant to the left-hand side. So when you do that, we have 2X minus 1 equals the square root of 5. Yeah, we've gotten what we wanted. Yeah, because we need to remove this radical sign. Now that it is standing alone, we can square both sides of the equation to eliminate it. So, squaring both sides of the equation, we have (2X minus 1) squared equal to the square root of 5 squared. Now, when you have A minus B squared, it is equal to A squared minus 2AB plus B squared. In this case, our A is 2X and our B is 1. Alright, so we are going to express these brackets in this form. So to do that, we're going to have 2x squared, right? A squared minus 2AB. 2 times 2X times 1 plus B squared. That is 1 squared. That's all for this side. Is equal to this, we definitely cancel this and we have 5. So if we solve this, we're going to have 2 squared is 4 and X squared is X squared. Minus 2 times 2X gives us minus 4X plus 1 equals 5. Now I will try to make X squared the subject of the formula. As I said, I'll explain why after a few steps. Now that I have made X squared the subject of the formula, I'm going to tell you why I'm doing that. So we are going to have, because I want to make X squared the subject of the formula Let me move everything connected to X squared to the other side. So I will have 4x squared equal to 5. Now minus 4X will cross over to become plus 4X. And plus 1 will carry over to become minus 1. Alright, if this is the first time you've seen this amazing channel, Click the subscription button and turn on your notification bell so you don't miss our amazing videos. We upload videos every day. So tune in at your convenience to watch for new uploads or to check for new uploads or to see what's new. And please give us a thumbs up if you're actually enjoying this video. Thank you so much. Now this is where we are. So we can say 4X squared is equal to 5 minus 1, which gives us 4 plus 4X. Now, to get the value of X squared, we are going to divide both sides by 4. So we have divided by 4, divided by 4, divided by 4. So we have X squared is equal to 1 plus 4X. Now this is a very important equation in this particular video. Now it's time to explain why I am doing what I am doing. Now recall that we said let our X be equal to 1 plus the square root of 5 over 2. So that simply means that what we are trying to simplify is X to the power of 6, right? Now I'm going to show you why I'm having trouble getting X squared. Using the law of indices or exponents, X to the power of 6 can be expressed as X to the power of 3 raised to the power of 2. So it simply means that if I'm able to raise my X to the power of 3 and then square it, I will get X to the power of 6, which is what we are asked to simplify. Can you see that? Yes. How do I get X to the power of 3 from X to the power of 2? Recall that A to the power of M plus N is equal to A to the power of M multiplied by A to the power of N. Therefore, since I have X squared, to obtain X cubed from here, All I need is to add 1 to it, right? That means I can say X squared, which is M multiplied by another X to the first power. So if I should multiply both sides of this equation by X to the power of 1, I will definitely get X to the power of 3, which I will square to obtain what we are asked to solve. Amazing, right? Yeah, I told you you're going to enjoy this. Now let's go. So since this is what we have, we have X squared equals 1 plus X. Now let's multiply both sides of the equation by X. Remember why? So we are going to have X multiplied by X squared equal to X multiplied by 1 plus X. Alright? So we are going to have X to the power of 1 plus 2 equal to X times 1, which is X. and X times X is what? x squared So we now have X cubed equal to X plus X squared. Now, whenever you're solving anything, okay, and you come across AFTER X squared, Simplify it by substituting this back. That will help you simplify your equation without making it look complicated. Alright? So I'm going to replace X squared with 1 plus X immediately. So I'm going to have X cubed equal to X plus X plus 1, or 1 plus X. Anyone? Okay. So x to the power of 3. Sorry, let me continue from here. So we have X cubed equal to X plus X. We give ourselves 2X plus 1. Alright. Now, recall what I said. We have obtained X to the power of 3. So we need to square it to get X to the power of 6, which is what we are asked to simplify. Alright. So to square, we are going to square both sides of this equation to obtain that. So we have X cubed squared equal to 2X plus 1 squared. Amazing, right? Yes. So here we have X to the power of 6 is equal to. So when you have A plus B squared, it is equal to A squared plus 2AB plus B squared. Okay. 2X squared plus AB, right? 2AB. 2 times 2X times 1 plus B squared. That is plus 1 squared. Okay. So I will have X to the power of 6 is equal to 4X squared plus 4X plus 1. Alright. So X to the power of 6 is 4 parentheses. Whenever you see X squared, replace it with 1 plus X. So 1 plus X. Alright. Plus 4X plus 1. So we have X to the power of 6 is 4 times 1, which is 4. 4 times X is 4X plus 4X plus 1. Alright. Yeah. So we have X to the power of 6 is equal to 4 plus 1. We give us 5 plus 4X plus 4X. We give ourselves 8X. Now, it is time to introduce what we have. The original equation. Sorry. Recall that we called X equal to 1 plus 5 over 2 to simplify the equation. So this is the time to bring it back into the equation. We are going to replace this X with 1 plus the square root of 5 divided by 2. So we have 1 plus root 5 divided by 2 raised to the power of 6 is equal to 5 plus 8 bracket. 1 plus the square root of 5 divided by 2. Okay. I hope you're enjoying this. If you are, please give us a thumbs up, comment, like, and share as well. Alright. Okay. That simply means that 1 plus the square root of 5 divided by 2 raised to the 6th power equals 5 plus. So we divide this to get 4, right? So we have 4 times the quantity 1 plus the square root of 5. Yes. So we have that this is equal to 5 plus 4 times 1, which is 4, and 4 times root 5 is 4 root 5. Okay. So this is equal to 9 plus 4 root 5. And this is the simplest form of (1 plus the square root of 5) divided by 2, raised to the sixth power, without using a calculator. Thank you so much for watching, and goodbye.