Olympiad Mathematics Trick | Can You Do This? | No use of calculator.

Added: 2026-06-02

[00:00:00]Hi everyone.

[00:00:03]How do we simplify this without the use of calculator?

[00:00:09]60 to the power of 6 minus 50 to the power of 6 equals what?

[00:00:18]Okay. Um look at the power. You know, 6 is the same as 3 3 * 2, right?

[00:00:25]So, in place of the power 6 I will write 3 * 2.

[00:00:33]minus 50 to the power of 3 * 2 again.

[00:00:39]And what can I do? Remember that if you have a to the power of mn you know, this can be written as a to the power of m to the power of n. We can separate it this way. Since it is multiplication that exists between the power.

[00:01:01]So, we will now have 60 to the power of 3 squared minus 50 to the power of 3 and this will still be squared.

[00:01:15]Then you come down to our popular difference of two squares.

[00:01:21]That says a squared minus b squared will give us a minus b times a plus b.

[00:01:33]This is our difference of two squares.

[00:01:36]So, this will, you know, we're going to use this identity to simplify what we have.

[00:01:42]Our a now is going to be 60 to the power of 3, then minus there we have um 50.

[00:01:50]Okay, by the way, we don't have to have this again, right? So, we'll have 60 minus 50 multiplying 60.

[00:02:00]Okay, that's cube.

[00:02:02]That's cube.

[00:02:04]Okay, that is still cube.

[00:02:07]So, we have these two multiply 60 to the power of three, then plus 50 to the power of three.

[00:02:16]Okay, remember our A and I 60 cube and our B is 50 cube. So, I put them into the identity that we talked about before.

[00:02:26]Now, here we have difference of two cubes and here we have addition of two cubes. Now, let me show you the identity again.

[00:02:36]A cube minus B cube is the same thing as A minus B into bracket.

[00:02:46]We'll open another bracket. We have A squared plus AB plus B squared, right?

[00:02:54]Then, what if it is addition of two cubes? We'll have A cube plus what? B cube.

[00:03:02]And this identity is A plus B into A squared minus AB plus B squared.

[00:03:15]This is for the addition. So, what do I do?

[00:03:19]What do I do? So, for this, let me break it down and do it separately. For the 60 cube minus 50 cube this will now be equal to 60 minus 50. This is the difference, right?

[00:03:35]So, we are using this one.

[00:03:37]>> [snorts] >> 60 minus 50 into bracket, we have A squared, which would be 60 squared plus AB. Our A is 60 multiplied by B, which is 50, right? Then we have plus B squared and that is going to be 50 squared.

[00:04:01]Right? So, this one is for the difference of two squares. I will deal with this first and then come back to the other one.

[00:04:11]Okay, 60 minus 50 is 10.

[00:04:14]Open bracket, 60 squared, that will give us 3,600.

[00:04:22]Then plus 60 times 50 is 3,000.

[00:04:30]Then plus here we have 2,500.

[00:04:36]Okay, let's um simplify this.

[00:04:40]We have 10 to multiply 3,000. Okay, let's add from here. 0 plus 0 plus 0 plus Okay, 0 plus 0 plus 0, that is 0.

[00:04:53]Then we have another 0 again.

[00:04:55]Then we're going to this five. Five plus 0 plus six, that's going to be 11. So, we write one, take one. We go to this two.

[00:05:06]Two plus three, five.

[00:05:09]Five plus three, eight. Plus one and we have nine.

[00:05:15]Okay, so 10 times 9,100, that will give us 91,000 and that is 91,000.

[00:05:26]Right? We have 91,000.

[00:05:29]This 91,000 is for the difference of two squares. I'm going up there. It's for this part. So, we have to try this part immediately.

[00:05:42]Okay, so for this part, remember this is the same as 91,000.

[00:05:48]So we want to do this now.

[00:05:50]And um our 60 cubed plus 50 cubed will now be equal to 60 plus 50.

[00:06:03]Okay.

[00:06:04]Into the identity. Here we're going to have um 60 squared.

[00:06:12]Then we have minus 60 times 50.

[00:06:17]Then we have plus b squared, which is going to be um 50 squared.

[00:06:24]50 squared.

[00:06:26]So this is the identity. Sorry I wrote all of those out of sight.

[00:06:31]Okay, so this is it.

[00:06:33]And um from here we have our 60 plus 50, which is 110.

[00:06:41]Then it is multiplying 60 squared. We said it is 3,600.

[00:06:46]Then minus 60 times 50, that is 3,000.

[00:06:53]Then plus this is 2,500.

[00:06:58]So at the end of the day we have 110.

[00:07:02]And it's multiplying 3,600 minus this. This is going to be um 600, right?

[00:07:09]600 plus 2,500.

[00:07:13]So what do we have at the end of the day?

[00:07:16]The addition of this is going to give us um Let's add this part. We have 1 10 times >> [snorts] >> 0 plus 0 is 0. 0 plus 0 is 0. 5 [snorts] plus 6 is 11, and it will make this one to be three.

[00:07:33]So our answer now will be 110 multiply by 3110.

[00:07:43]We are not supposed to use calculator as we rightly said. So, we have to multiply. Let's do it together here.

[00:07:51]Okay, so we have 3100 * 110.

[00:07:58]This is multiplication.

[00:08:00]0000 This is 0013.

[00:08:07]Then we have 00 13.

[00:08:12]So, we add this. Sorry, I wrote out of sight again.

[00:08:17]The addition will give 0 0 0 1 We have 4 and 3.

[00:08:27]So, this is um 341,000.

[00:08:33]So Okay, so this means that here now we have 341,000.

[00:08:42]So, this is what we have. Now, we have to multiply this again.

[00:08:49]Okay, so from here to multiply this the fastest step to take is to factor out 1,000.

[00:08:58]So, we take 1,000 out.

[00:09:01]Then here we have 91.

[00:09:03]Then here we have 341.

[00:09:07]So, here now we're going to multiply 341 by 91. Let's do that quickly.

[00:09:14]This is 1, this is 4, this is 3.

[00:09:17]9 * 1 is 9. 9 * 4 is 36. We write 6 and we're taking Okay, sorry, I wrote out of sight. So, this is um 9 * 1 is 9, 9 * 4 is 36.

[00:09:31]We're taking 3. 9 * 3 is 27. 27 + 3 is 30.

[00:09:39]So, we have um one.

[00:09:42]This is three.

[00:09:43]Take one into make this one 10. This is one, and this is three.

[00:09:48]So, we have um 31,031.

[00:09:53]Okay. We have 31,031.

[00:09:56]So, let me remove this.

[00:09:59]So, this means that we have 1,000 multiplied by 31,031.

[00:10:09]Right? So, when we multiply these now, we're going to have 31 031, then add these three zeros, 1 2 3. 1 2 3, we have these 1 2 3.

[00:10:23]This is 31 million 31,000.

[00:10:27]So, what are we saying? We are saying that 60 to the power of 6 minus 50 to the power of 6 is equal to 31 million 31,000.

[00:10:42]Thank you for watching.