United States|Math Olympiad| can you solve this.

Hi, let's simplify this particular Olympiad question. So I want you to learn the trick, that whenever you see this type of question, This is how you approach it. Now, when you look at the numerator, just take this step. OK, and I'm going to give you assignments at the end. Something similar for you to try out yourself. So watch to the end and let's go. Now, when you look at the numerator, the lowest power we have there is 11. So we are going to manipulate all these numerators, all these powers, in terms of 11. That is the first thing we should do. Now, how do we manipulate this 33 in terms of 11? We can write 33 as 11 plus 22, right? So we write 2 to the power of 11 plus 22. This is the same thing as 2 to the power of 11 plus 11. plus 2 to the power of 11 divided by... I am going to solve about three different questions here. So just sit still, grab your popcorn, and keep picking it one piece at a time. So on this side, just do the multiplication here. What number multiplied by 11 gives you 33? What is it? 3. So we have 2 to the power of 11 multiplied by 3 minus 1. All right. So this is equal to... Now, recall that when you have a to the power of m, when you have a to the power of m plus n This is equal to a to the power of m multiplied by a to the power of n. Okay. All right. So this is equal to... That means we can write this as 2 raised to the power of 11 multiplied by 2 raised to the power of 22. 2 to the power of 11 multiplied by 2 to the power of 11, plus 2 plus 2 to the power of 11 Remember that in this type of question, you are not allowed to use a calculator. So we are going to simplify it as much as possible. Okay. Now here, when you have a to the power of m, n. Okay. This is the same as a to the power of m raised to the power of n. Okay. a to the power of n raised to the power of m. Any of them. Okay. Now, let's go back to this. That means I can rewrite this, okay, as this denominator. Yeah, this denominator. That means I can rewrite it as 2 raised to the power of 11, and then that whole thing raised to the power of 3, right? -1 Okay. Now let's continue. So when you look at this, when you look at the numerator, Notice that we can factor out 2 to the power of 11. That is exactly what we will do. So this is equal to 2 to the power of 11. We open our bracket. When this is divided by this, what do we get? 2 to the power of 22 remaining. Moreover, when this is divided by this, we are left with 2 raised to the power of 11. Moreover, when this is divided by itself, we get 1. Divided by 2 to the power of 11 raised to the power of 3 minus. That 1 is the same thing as 1 to the power of 3, right? Yes, you see why I had to add that 3. Now we are going to simplify the denominator. So when you have a to the power of 3 minus b to the power of 3, This is equal to (a minus b) multiplied by (a squared plus ab plus b squared). Okay? a squared plus ab plus b squared Now we are going to substitute a = 2¹¹ and b = 1. Therefore, we can rewrite this as what? 2 to the power of 11 parentheses. 2 to the power of 22 plus 2 to the power of 11 plus 1. Everything divided by. Now let's simplify the denominator. Okay? So the denominator is the same as 2 to the power of 11 minus 1, right? Multiplied by a squared. That is 2 to the power of 11 squared. Okay? Plus ab. That is 2 to the power of 11 plus. Sorry. Multiplied by 1. Plus b squared. That is 1 squared. Okay? This is what you will have. Now this is equal to 2 to the power of 11, enclosed in parentheses. 2 to the power of 22 plus 2 to the power of 11 plus 1 divided by So we have 2 to the power of 11 minus 1 open bracket. That's for this one right here. So when you have two exponents on a number, you can multiply them together, right? So we have 2 to the power of 11 times 2, which gives us 22. Alright. Adding 2 to the 11th power times 1 still results in 2 to the 11th power. Plus 1 squared is still 1. So this is equal to. Now, if you look at this thing very carefully, notice that this one and this one are the same. So they will cancel each other out. Now we have 2 to the power of 11 divided by 2 to the power of 11 minus 1. Now we continue. So this is equal to 2 to the power of 10 plus 1, divided by 2 to the power of 10 plus 1, minus 1. Okay? So this is equal to 2 raised to the power of 10 multiplied by 2 raised to the power of 1, according to the law of exponents. Okay? Alright. Now divide that by 2 to the power of 10 multiplied by 2 to the power of 1 minus 1. Alright. Let me do this. To use this side of the whole thing. Excuse me. Let me draw my line. So please, if this is the first time you're seeing us, what's still holding you back from clicking that subscribe button? Please click the subscription button. Turn on your notification bell. If this is the first time you're seeing us, so you don't miss our amazing videos. And please give us a thumbs up if you're actually enjoying the video, which I'm sure you are. So let's go. Now we have, this is where we are. 2 to the power of 10. Okay? Multiplied by 2 to the power of 1. Divided by 2 to the power of 10. Multiplied by 2 to the power of 1 minus 1. So this is where we are. Now we can easily calculate 2 to the power of 10. Alright? 2 to the power of 10 is 1024. So we have 1002. Since I've already written two, let me use this one as the second. 2 to the power of 1 is 2. Multiplied by 1024. Right? That is 2 to the power of 10. Divided by, now 2 to the power of 10 is 1024. Multiplied by 2 as well. Multiplied by 2. -1 Now this is equal to. When these multiply, they are going to have 4048. Right? 4048 divided by here. We are going to have 4048 minus 1. Okay? So when we solve this further, we can see that this is equal to 4048 divided by 4047. So I think this is the simplest so far. Alright. Thank you so much for watching. I would have loved to give you an assignment, but unfortunately, I'm sorry. In my next video, I'm going to do that. Okay? So watch my next video for another interesting trick from the Olympiad, where I'll show you something. to solve problems related to what I just taught Okay? Thank you for watching. Goodbye.