A trending math Olympiad radical/ surdic question.

around and see how we are going to solve this is actually very interesting now we are asked to In this type of question, there is only one piece of knowledge. There are some pieces of knowledge you need to know, but let's just start with the basic one; the first one is that you know that when you have for example one over two divided by one over two okay it's the same thing as one over two times that is, if you change your division to multiplication, you are going to get what They are going to take the inverse of the other number, so it's going to be times two over one. that means one over two divided by one over two will give you one, all right, so the essence is that When you see division, you turn it into multiplication, but when you turn any division For multiplication, you have to take the inverse of any number or any fraction following it. division sign, so we are going to apply exactly the same thing here, all right? So applying that. same thing here we are going to write that this is equal to the square root of one twenty-five one over Twenty-five times once, when I put this multiplication, I'm going to change everything here towards 25 over one, all right, so it's going to be 25 over one times 25 over one times 25 over one. times 25 over one, okay, so we go ahead and divide this by this, and we are going to have 25 times 25 times 25, so this is equal to the square root of 25 times 25 times 25. Okay, note that this It's the same thing as the square root of 25 to the power of 3, right? Yeah, it's the same thing as 25. to the power of 3 now note and remember that in indices square root of a is the same as a to the power of one half a to the power of half times the square root of a is the same as a to the power of half, guys. If I I want to remove this square root sign, so I'm going to raise a to the power of one half, in the same manner. in the same way, relating this to this, that means that if I want to write this 25 to the power of The square root of 25 to the power of 3; that's all I need to do is to do what 25 to the power of 3. raised to the power of what half okay now remember also that a to the power of b raising to the power of c is the same thing as a to the power of b times c Alright, yeah, that's when you have two powers on a particular number—you multiply them out, just like that. multiply the two powers out so that you end up with only one power for the number, which is exactly what we are going to do it now, but before that, if this is the first time you are tuning into this channel Please don't hesitate to subscribe to the channel, then um, hit that notification bell. so that you'll be notified whenever I upload a new video, and also don't hesitate to like Comment, watch, share. Thank you so much. So we are going to apply what I just explained to this. That means this is the same as 25 to the power of 3 times half, which gives you 3 over 2. Right, okay, there is another law of indices, and that law says that a to the power of x, okay, let's x to the power of a over b is the same as the bth root of x raised to the power of a. Okay, the bth root of x raised to the power of a, so we are going to apply the same thing here. Here, it means that 25 to the power of 3 over 2 is the same as the square root of 25. raised to the power of 3, all right, so this is equal to the square root of 25, which gives us 5. to the power of 3, now this same thing as 5 times 5 times 5, which will give us 125 And this is the solution. Thank you so much for watching, and do have a great day. Bye.