Olympiad Mathematics Trick | No use of calculator | Can you do this?

[00:00:00]Everyone, if you're ready, let's simplify this without the use of calculator.

[00:00:12]Square t of 24 to the power of 5 + 20 to the power of 5. So how do we simplify this without using calculator at any point?

[00:00:29]So we have to understand that 24 is the same thing as 2 * 2 * 2 * 3 and that is 2 ^ 3 * 3.

[00:00:46]This is the same thing as 24, right?

[00:00:48]Then let's bring out that 20 again.

[00:00:52]20 is um 2 * 2 * 5 right this is 20 and it's 2 to the^ of 2 * 5. So in place of 20 and 24 we're going to put these values.

[00:01:12]Now we have the square t of 2 ^ 3 * 3 this is raised to power 5 plus in place of 20 we are putting 2 ^ 2 * 5 and this combined is raised to the power of five as well.

[00:01:37]So what do you think I'm going to do from here?

[00:01:41]Um let's do something.

[00:01:44]Do you know that a b to the power of c is the same thing as a to the power c * b to power c.

[00:01:56]Okay, this is from one of the laws of um indices that we know.

[00:02:02]So now applying what I just explained we're going to have 2 ^ 3 * 5 then mult* 3 ^ 5 this is from here then from this part we have 2 ^ 2 * 5 then we have um what do we have we have 5 to the power of five over there. So see what we are going to do. This is the square root of 2 to the power of 15.

[00:02:42]2 to the power of 15 * 3 to the power of 5 + here we have 2 to the power of 10 * 5 to the power of 5.

[00:02:58]The next point is that we bring out a common factor. You know at this point we do not have any common factor.

[00:03:07]Right? But we know that we can express this as 2 ^ 10 + 5. You know that will give us 15 as well. Then multiply by 3 ^ 5. Then we have + 2 ^ 10 * 5 ^ 5.

[00:03:31]Then the next point is that we use this um law a to the power of m + c is the same thing as a ^ m * a power c. Right?

[00:03:45]So what we have here will now exist in this form.

[00:03:50]So that we can now have the square t of 2 ^ of 10 * 2 to the power of 5 * 3 ^ 5 + 2 ^ 10 * 5 ^ 5.

[00:04:13]Okay. So from here now we have a common factor already and that is 2 ^ 10 open bracket here we have 2 ^ 5 remaining times there we have um 3 to power 5 as well then plus there this is already out so we have this one remaining and that is 5 to power what five okay so what again do you think we are going to do I think we should combine this right let's combine this and this is happening from one of the laws that says that if you have a ^ m * b to power m this can be combined to give a b both to the power of what m so if I apply that to what we have we are going to have 2 to power 10 into we have um 2 ^ 5. Okay, let's combine it.

[00:05:15]So we have 2 * 3, right? So both of them this we now have the same power of five.

[00:05:34]Then we yeah this is okay. Then we have + 5 to the^ of 5. Then we close it.

[00:05:44]Right?

[00:05:45]Yes, this is it. Then our expression will now be 2 ^ 10 into bracket 2 * 3 is 6. So we have 6 ^ 5. Then we have plus what? 5 ^ 5.

[00:06:05]5 to the^ of 5.

[00:06:10]Okay. So we are going to continue from here as we have the square root of 2 to power 10 into what is 6 ^ 5. First of all 6 ^ 4 is 1,296.

[00:06:32]So let's multiply by um what do we do? multiply by six, right?

[00:06:40]Okay. So, we'll multiply this by 6 so that we can have 6 to the power of what?

[00:06:48]Can have 6 to the power of 6. Now, 6 * 6 is going to give us 36 and we're taking um 3. Then 6 * 9 is going to give us um 54, right?

[00:07:06]So that 54 + 3 that we are carrying it will give us um 57. So we write seven sorry we write seven.

[00:07:18]We write seven and we are taking five.

[00:07:23]Then we go on to get 6 * 6 is 12. 12 + 5 is 17. We are taking 1. 6 * 1 is 6 + 1 is 7 is 7 is um it's 7. Yeah, it's 7. So that means we're having 7,776.

[00:07:46]So this is what we have here. 7,776 plus let's look at 5 to the power of 5.

[00:07:58]Okay. Um 5 to^ 5. I know that 5 to^ 4 is 625.

[00:08:05]So we multiply by five and that will be 5 to power 5. This is 25. 5 take what?

[00:08:11]2. 5 * 2 is 10 + 2 12 take 1. This is 30 + um 131. So we have 3,25.

[00:08:26]So here we have um 3,125.

[00:08:32]So we proceed, right?

[00:08:36]And we're going to have the square root of 2 to the^ of 10 into what is the sum of this? Let's add from here. 5 + 6 that is 11. Take 1. 2 + 7 that is 9 + 1 10 and we have to take 1 again. 1 + 7 that is 8 + 1 that is 9. Then [snorts] we have 3 + 7 that is um 10.

[00:09:11]So this is um 10, 91.

[00:09:18]Okay. So the next step is that from what we have here we can break into two to get the square root of 2 ^ 10 multiply by the square t of 10,91.

[00:09:33]Now from here the square root will divide this power to give us five. So we have 2 to^ 5 then we are multiplying by the square root of 10,91.

[00:09:50]Okay. So this means that we have 2 ^ 5 is going to be 32. So we have 32 square root of 10,91.

[00:10:02]And um apart from one there's no other perfect square that is a factor of 10,91.

[00:10:10]So what are we now saying? We are saying that the square root of 24 to power 5 + 20 to power 5 is equal to 32 root 10,91.

[00:10:33]Thank you for watching.