Olympiad Mathematics Trick | Indian Can You Simplify This?

[00:00:01]Hi everyone, how do you simplify this without the use of calculator?

[00:00:09]Okay, we have 64 to the^ of 4 - 60 to the power of 4.

[00:00:18]Um this is simple.

[00:00:21]64 is the same thing as 2 to the power of um six. So this is to the power of four. Then 60. Okay, let's do 60 here.

[00:00:34]Two here we have 30.

[00:00:38]2 again we have 15. Then we have three.

[00:00:42]Five we have 5 1. So 2^ 2 * 3 * 5.

[00:00:48]Right? So let me remove this.

[00:00:53]So in place of 60 we put 2 squar okay 2 * 3 * 5 so everything here is to the power of 4 let me extend this so what do I do remember that what we have here can be expressed in a different form because we know that if you have A B C and both of them are raised to the power of D. This is the same thing as A to the power D * * B to power C * C everything is to power of D * C to the power of D. So this will help us simplify further as we have the square root of 2 ^ 6 to power 4 - 2 ^ 2 then this will have power 4 then we have * 3 to power 4 again then * 5 to the same power four.

[00:02:03]So from here something has to be done because we know that if you have a to power b to power c this is the same thing as a to the power of b * c. So we are to always multiply the powers. So we having square root of 2 to the power of um what do we do?

[00:02:26]Um 6 * 4 is 24. So we have 2 to the^ of 24 minus here we have 2 to the^ of 8 minus okay times * 3 ^ 4 * 5 ^ 4. So to continue with what we have we are going to try to have something like um common something common. So we are going to have 2 to the^ of 24 can be written as 2 ^ 8 + 16. Then we have - 2 ^ 8 * 3 ^ 4 * 5 ^ 4. Remember 8 + 16 is 24.

[00:03:16]So from here now we'll apply one of the laws of indices. This is the same as 2 ^ 8 * 2 ^ 16.

[00:03:26]Okay? Because if you pick one of the bases, you're going to have to add the powers. - 2 ^ 8 * 3 ^ 4 * 5 ^ of 4. So guess what I'm going to do? I already have a common factor. What is a common factor?

[00:03:49]2 to the^ of 8 is a common factor. So what will be remaining in the bracket is 2 ^ 16 from here then minus there we have 3 to power 4 then multiply by there what do we have? We have um five to the power of 4.

[00:04:14]Okay. So from this point what do we do?

[00:04:18]Let's still go on.

[00:04:22]We have um the square root of 2 to the^ of 8 into now 2 ^ 16. I think it's going to be very large. Let's break it. We're going to have 2 ^ 8 + 8 - 3 ^ 4 um * 5 ^ 4 So if we go further we are going to have the square root of 2 to the^ of 8 and then here we're going to have 2 ^ 8 * 2 ^ 8us what is 3 ^ 8 sorry 3 ^ 4 that is 81 * 5 ^ 4 that's going to be 25 * 25 and is 625.

[00:05:19]Close it.

[00:05:21]Right. So what do we do next? We have the square root of 2 to power 8. Then this is um 2 ^ 8 is 256.

[00:05:34]So we have 256 * 256 - 81 * 625.

[00:05:43]So what will this give us?

[00:05:47]Um let's let's proceed. Let's pro this.

[00:05:50]Remember we don't use calculator, right?

[00:05:53]So this is going to take us um a little time to multiply. We have 256 * 256.

[00:06:02]Let's multiply. 6 * 6 is what? We have 36. So I'm going to write 6. Take 3.

[00:06:09]Then 6 * 5 is 30 + 3 33. 6 * 2 is 12 + 3 that will be 15. Then 5 * 6 is 30 take 3. 5 * 5 is 25 + 3 28 take 2. 5 * 2 is 10 + 2 that will be 12. Then 2 * 6 is 12. Take 1. 2 * 5 is 10 + 1 11. Take one. 2 * 2 is 4 + 1 that is 5. Then we have the addition of everything. 6 comes here. 3 comes here. This is 15. We are taking one.

[00:06:53]Then this will turn to five and this is 6. So we have 65,536.

[00:07:02]65,536.

[00:07:05]So let me remove this.

[00:07:08]So here now we have our square root of 2 to the^ of 8 into 65,536.

[00:07:20]Then minus we have the product of this.

[00:07:22]Let's multiply that as well. [snorts] We have um the bigger one should come first. 625 * 81. So this time this is five. Here we have two. Here we have 6. Then this time this is 40. We take four. This is 16 + 4 that's going to be 20. We're taking two.

[00:07:45]Then we have um this multiply by this is 48 right?

[00:07:52]48 + um + 2 that's going to be 50. So the addition is five.

[00:08:03]Okay. So the addition is going to be five. We have two. We have six. We have zero. We have five. So we have 50,625.

[00:08:16]Interesting, right? So here we have 50,625.

[00:08:23]So we close this and then we go again.

[00:08:26]We have the square root of 2 to power 8 into we subtract this two. I think we can do that. Um 6 - 5 that's going to be 1. 3 - 2 that's going to be 1. Then 5 - 6. Take one to make it 15. 15 - 6 is going to be 9. So this one becomes 4. 4 - nothing is 4. Then we have um 6 here.

[00:08:56]Right? So 6 - 5 that is going to be 1.

[00:09:01]So we have um 14,911.

[00:09:07]So from this part we can split what we have now and we're going to have okay look at the splitting we have 2 to the^ of 8 * the square root of 1 4 91 1. So from this point I think we are almost done. So this one will go into this how many times? Two uh it will give us four right. So we have two to the^ of 4 multiplying the square root of 14 911 1 and this will eventually give us 2 ^ 4 is 16. So this is 16<unk> 14 911 4 911. So this is it right. So to conclude 64 to the^ of 4 I mean the roo<unk> of 64 to the^ 4 - 60 to the^ 4 is equal to 16<unk> 14,911.

[00:10:19]Thank you for watching. You can always confirm this yourself.