Hardest question|99%failed|Germany| A Nice math Olympiad exponential simplification.

So 99% of the students to whom this question was posed did not get it right. I don't know why, but let's get it right. Now, we are told to simplify this or to evaluate this. First, to simplify it, let's call this particular number inside the parentheses something. So let's take 1 plus the square root of 5, divided by 2, and let's call it y. Or let's call it something; I don't want something that is long, let's call it a. Yeah, this one is smaller. So if we say let this be equal to a, let's cross-multiply. So if we cross-multiply, we will have 2 times a, which gives us 2a. Is equal to 1 plus what? Square root of 5. Now, let's move this to the left-hand side of this equation. If we do that, then 2a minus 1 will be equal to the square root of 5. Alright, so in order to get a number here, let's take the square root of both sides. Then square both sides so that we can eliminate this radical sign. Alright, so if we say (2a minus 1) squared, this should be equal to the square root of 5 squared. So we are going to have that 2a minus 1 squared is equal to what? 5 because this square will cancel out the square root. Alright, now this is what we have because I want to change position That (2a − 1)² equals 5. Perfect. And we continue now. Recall that a minus b squared is equal to a squared minus 2ab plus b squared. In this case, our a is 2a and our b is 1. So we are going to relate this to this. So to do that, that means this is going to be squared. That is 2a squared minus 2 times a. Remember, our a is 2a, right? Multiplied by b, which is what? 1. Plus b squared, which is 1 squared, is equal to 5. Alright, now let's work this out. So we have 2a squared, so 2 squared gives us 4. a squared is equal to a squared minus 2 times 2a, which gives us 4a plus 1, and this is equal to 5. Now let's bring this in. If we do that, we are going to have 4a squared minus 4a. Then this will go inside to become 1 minus 5 equals 0. Now let me explain what I'm trying to do here. We're told to raise this to the power of 12 and call it a, right? You know that a to the power of 12 is the same thing as a to the power of 6 squared, right? That means if we are able to raise a to the power of 6 and then square it, We are going to solve this problem, right? So all we are going to do here is try to get a to the power of 6. And a to the power of 6 is the same thing as a to the power of 3 squared. So we are going to emphasize or we are going to try to achieve these two things. a cubed and a to the sixth power Once we get a to the power of 3, we square it. Then we get a to the power of 6. What to do? We square it as well. Simple. So in all of this, one thing is very important. I'm going to take a look at that. So I would like you to stick to the end and see how this plays out because it is a very interesting mathematical problem. Alright, so we have 4a squared minus 4a minus 4 equals 0. 4, 4, 4, 4, 4. Let's just divide through by 4. If we do that, a squared minus a minus 1 equals 0. Now let's make the square the subject of the formula. Remember what I showed you—what we're looking for, right? So we have a squared equal to minus a, we cross to become plus a. and minus 1 we carry over to become plus 1. And this is a very important equation. Let's call this equation (1). Because anything we are trying to achieve, We are trying to get a to the power of 3 and we are trying to get a to the power of 6. You must meet at the power of 2 there. So we need to simplify them. Now we have obtained a to the power of 2. It's easy to get a to the power of 3. If we multiply through by a, we are going to get a cubed. So let's go ahead and do that. If we multiply through by a, we have a squared multiplied by a to the power of 1. is equal to a multiplied by a plus 1 multiplied by a. a squared multiplied by a to the power of 1. Recall that a to the power of m multiplied by a to the power of n According to the law of exponents, a to the power of m times a to the power of n is equal to a to the power of m plus n. Right? That's why I said that when we multiply through, we're going to get a to the power of 3. Because this becomes a to the power of 2 plus 1, which is 3. is equal to a times a, which is a squared, plus what? A. Perfect. C squared. I told you this is a very golden equation. So this particular a squared is really important to what we are doing. Now let's continue. Let us continue. Okay, if this is the first time you're coming across this channel, What is still preventing you from clicking the subscribe button? Please click before you forget. Subscribe to the channel, turn on your notification bell so that you will be notified whenever I upload a new video. And you know what? Please try to comment, like, and share. That will help the algorithm show my videos to a wider audience. And thus, it will reach those who need it more. Okay? Thank you so much. Now take a look at what we have here. So we have a to the power of 3 is equal to. Now this is squared, we are going to replace it with this. Yes. So we are now going to write a plus 1 plus a. You can see how it simplified this particular equation. So we now have that a to the power of 3 is equal to a plus a will give us 2a plus 1. A very important equation. Again. Perfect. Now recall what we are aiming for. We are aiming at a to the power of 3 to square it to get a to the power of 6. To square a to the sixth power to obtain our solution. So we are halfway to our solution. Congratulations if you watched all the way to the end. Now let's continue. So we are going to square a to the power of 3. a to the power of 3 squared Is equal to a to the power of 6, right? Yes, because when you have a to the power of m and this is also raised to the power of n according to the law of indices or exponents This is equal to a to the power of m times n. Alright? Okay. Now let's continue. So we have this as a to the power of 3. So we are going to have a to the power of 6. is equal to what? a cubed Okay, let me not repeat what I've written here. Let's just dive in. So this is a to the power of 3, which is 2a plus 1. raised to the power of 2. So a to the power of 6 is equal to so we have a plus b squared In algebra, it is equal to a squared. a squared plus b squared plus 2ab a squared plus 2ab plus b squared. Anyone is still correct. Now, we are going to substitute whatever we have into this. Alright? Now, this implies that this is squared, right? That is 2a squared, which will give us 4a squared. So we have 4a squared here plus 2ab, right? That is 2 times 2a, which is 4a. times 1, which is still 4a plus 1 squared That is b squared, which is 1. So we have that a to the power of 6 is equal to 4. Wherever you see a square, You have to replace our lovely equation. So we are going to substitute a plus 1 here to simplify this. Do you understand now? Okay. So we are going to add a plus 1 here. Alright? Sorry, not squared again. a squared is a plus 1. Alright, plus 4a plus 1. a to the power of 6 is equal to 4 times a will give us 4A and 4 times 1 will give us plus 4. Alright? Plus 4a plus 1. a to the power of 6 is equal to 4a plus 4a will give us 8a. And 4 plus 1 will give us 5. Amazing! Yes! What will happen? We're almost there. Once we have a to the power of 6, We just go ahead and square it to get our solution. So let's do that together. And let's start from here. So, this is what we have here. That a to the power of... Okay, let me not repeat what we have there. a to the power of 12 is equal to a to the power of 6 squared. Right? And what do we have for a to the power of 6? We have 8a plus 5. Alright? So we have 8a plus 5 squared. So we go ahead and use our algebra to open this up So we have a to the power of 12 is equal to 8a squared. Right? Which is 16. 4a squared plus 2ab. That is 2 times 8a. That will give us 16a. Right? Times 5. That will give us 80. Right? Or let me do it this way. 8 times 5 is already 40. And 40 multiplied by 2 is 80. So this should be 80a. Then plus what? b squared That is 5 squared, which is 25. I just hope I haven't confused you here. And I think I didn't. All I did was that a plus b squared is equal to a squared plus 2ab 2ab plus b squared. Okay? So substitute your a with 8a and your b is 5. And you're going to get this. Alright. Let's continue. So we have a to the power of 12 is equal to 64. Recall what a squared is? 1 plus a So always write that. I hope I am right. Right? Yeah. a plus 1 or 1 plus a Alright. Plus 80a plus 25. Okay. So we continue. Now, Let's solve. Looking at this now, You can see that a to the power of 12 is equal to 64 times 1 will give us 64. And 64 times a will give us 64a. Alright? Plus 80a plus 25. We are still here. So we're still going to do something about it. We don't need to have an inside in this place. But don't worry. We're going to do that. So we have that a to the power of 12 is equal to 64 plus 25 What will they give us? We have 5 plus 4 is 9. 6 plus 2 is 8. So we have 89. 80 plus 64 will give us 144a. Right? Now, we have one here. This one is a. This is 9. So we have one here. But recall that our a is equal to 1 plus the square root of 5 divided by 2 Okay? So we are going to substitute this to this particular place. So we have a to the power of 12 is equal to 89 plus 144 multiplied by 1 plus the square root of 5 divided by 2 Now, this 2 is going to divide this to give us 72. Right? So we have a to the power of 12 is equal to 89 plus 72 bracket 1 plus root 5 Right? So we have a to the power of 12 is equal to 89 plus 72 equals 161. Times 1 will give us 72 plus 72 root 5. Okay? a to the power of 12 then should be equal to 89 plus 72 equals 161. 9 plus 2 is 11. We write down 1 and carry 1. Now, what do we get when we add 8 and 8? 16. So we have 161 plus 72 root 5. And this is our answer. All you need to do is I don't have enough space again. Just substitute a here. That is 1 plus the square root of 5 over 2 raised to the power of 12 Which is the question? is equal to this. And that's all. Thank you so much for watching And have a lovely day. Goodbye.