A Nice math Olympiad exponential question| can you solve for x?

Let's solve this, but we are asked to evaluate or simplify it. Okay, let's simplify it. Okay, we are going to let everything inside be called x, so we are going to write let The square root of 15 minus the square root of 3 divided by the square root of 12 equals x. Now this is a sword, so what we are going to do is simplify the sword first. Now there is a The law in mathematics that states that the square root of a times b is equal to the square root of a multiplied by the square root of b. the square root of b, okay, so that means I can bring out two factors of this and separate them Okay, so I can write √15 as √3 multiplied by √5, right? So I can say That means that root 3 multiplied by root 5 minus root 3 divided by this can be written as root 3. multiplied by the square root of 4, right now everything equals x, okay? Before I go further, please, I'm I'm not left-handed, so I'm just trying to write this properly, so just manage my writing, okay? All right. Now when you look at this, we can factor out √3 from the numerator, so we can write the square root of 3, brackets; when this is divided by this, this we can see, and we are left with So we have √5 minus, when √3 is divided by itself, we get 1, okay, all right. divided by, then we have root 3 multiplied by square root of 4, which is 2, and this is equal to Okay, now we have √3 here and we have √3 here, so they can cancel each other out. means what we have left is what that square root of 5, sorry, the square root of 5 minus 1 Divided by 2 is equal to x. All right, now look at this: we can do this. This is over 2. Then let's cross-multiply. When we cross-multiply, we have that 2x is equal to the square root of 5. minus what 1 okay then we write 2x let's bring this to this side to get plus 1 is equal to root 5 I want you to trust me, and we need to reach a point before I explain what I'm trying to do. Okay, all right. Now let us square both sides of this equation. Let's square both sides of So the equation is: (2x + 1)² equals the square root of 5. All squared, okay. Now what do we have when you have a plus b squared is equal to? a squared plus 2ab plus b squared, okay. All right, now in this case, our a is 2x and our b is 1, so What we are going to do is express this in this form, so we are going to write a that is 2x squared, all right, plus 2ab, that is 2 multiplied by 2x multiplied by 1, plus b squared that is 1 squared and everything is equal to now this we divide cancel this and we are left with 5 so 2 squared gives us 4, x squared gives us x squared, and 2 times 2x gives us 4x plus 1 is equal to what 5; now let's bring this to this side, we have 4x squared plus 4x plus 1 minus 5 equals 0, so we have 4x squared plus 4x minus 4 equals 0, okay this If this is the first time you're encountering this channel, please click the subscription button. and on your notification bell so that you'll be notified anytime I upload a new video, please Don't forget to give us a thumbs up if you're actually enjoying this video. Thank you, so I'm going to rewrite this last line so we have that 4x squared plus 4x minus 4 is equal to 0, so let's Divide through by 4, as I said—I'm going to actually explain what I'm doing at this point. I'm already getting there, so let's divide through by 4 to simplify this, okay? So this, we divide this. We have x squared plus x minus 1 equals 0. Now let's make x squared the subject of the formula. So we have x squared equals this plus x; we move to become minus x, okay, and minus 1 we cross to become plus 1, so we have x squared is equal to 1 minus x. Now this is a very important equation. Now let me explain: we are asked to solve this to the power of 6. Okay, this to the power of of 6, and we said let this be equal to x, right? For us to get x to the power of 6, we are going to use our law of indices or exponents, which states that a to the power of m raised to the power of n is equal to a to the power of m times n. Now we have a to the power; we have x to the power of 6, right? Yes, we have x to the power of 6. Now, to get x to the power of 6, we can get it by saying x to the power of 3 raised to the power of 2. So, if we are able to get x to the power of of 3 we will square it to get x to the power of 6, which is what we are looking for, all right, so That's why I tried to get x squared, because from this I can obtain x to the power of 3. if I multiply x squared by x, I'll get x cubed, okay, yes, because according to law of indices or exponents again when you have a to the power of m okay multiplied by a to the power of n, this is the same thing as a to the power of m plus n. So if I say x squared multiplied by x to the power of 1 I will get x to the power of 2 plus 1 which will give me x to the power of So now I will square my x to the power of 3 to reach my destination. You see why, all right? So let's go ahead and do that now we are going to multiply this by x to get x to the power of 3 so we are going to have x squared multiplied by x is equal to x multiplied by 1 minus x okay so we have x to The power of 3, just as I've explained so many times: x times x gives us x squared, and x times minus x gives us minus x squared. x squared, okay, so we have x to the power of 3 is equal to x minus. Anywhere we see x squared, we are We're going to replace it with this to simplify the equation, so we're going to have 1 minus x, okay. so we have x to the power of 3 is equal to x minus times 1 will give us minus 1 minus times minus x Okay, permits me to clean this side of the board. Okay, now we have x to the power of 3 is equal to x plus x will give us 2x minus 1. Now recall what I said—let's square this to get to x to the power of 6, okay, so x to the power of 3 raised to the power of 2 is equal to 2x minus 1 raised to the power of what 2 now we have x to the power of 6 3 times 2 is what 6 is equal to when you have a minus b squared, okay, this is the same thing as a squared minus 2ab plus b squared; it's the same So we are going to express this in that form, so it's going to be a squared, which is 2x. which will give us 4x squared minus 2ab, which will give us 4x plus b squared, which will give US 1, right? This is our A and this is our B. Okay, is this what we have now? We have that X to The power of 6 is equal to 4 brackets, so anywhere we see x squared, we put 1 minus x; yes, that is our x squared, right? I hope I'm correct because I've actually erased it. Okay, now minus 4x plus 1 Now x to the power of 6 is equal to 4; thus, we have 4. 4 times minus 6 will give us minus. 4x minus 4x plus 1, so we have x to the power of 6 is 4 plus 1, which will give us 5 minus 4x minus 4x, which will give us minus 8x. Now this is the time to do our substitution. Record that we said Let √15 − √3 divided by √5 be equal to divided by √12. be equal to x, but instead of that, I'm going to take this simplified version. Yes, we have simplify it here to get root 5 minus 1 divided by 2 as our x, so we are going to substitute for Here and for here as well, all right, so that's what I'm going to do right now. So we are going to write, but our x is equal to the square root of 5 minus 1, divided by 2. Okay, so let's substitute this is x to the power of 6 that is the fifth root of 5 minus 1, let me not use it here, let's just x to the power of 6 is equal to 5 minus 8; then let's use this root 5 minus 1. divided by 2, okay, so we have x to the power of 6 is equal to 5 minus this; we divide this to give minus 4 times the square root of 5 minus 1, so x to the power of 6 equals 5 minus minus 4 root 5 minus 4 root 5 minus 4 times minus 1 will give us plus 4, so x to the power of 6 is 5 plus 4 will give us 9 minus 4 root 5. Now the x I'm going to put here now is going to be this one. Yes, because this is the question; although I use the simplified form, why solve it here? If I don't use this, I'll keep simplifying it again, and I don't want to do that. So we can now say that x to the power of 6, that is x raised to the power of, sorry that is the square root of 15 minus the square root of 3 divided by the square root of 12, everything raised to the power of 6 is equal to our final answer; here is what 9 minus 4 root 5, and that is the answer I hope you enjoyed this video, please don't forget to give this a thumbs up if You actually did okay, thanks so much for watching, and have a lovely day.