Harvard University Test | No use of calculator | Olympiad Math Trick

Added: 2026-05-20

[00:00:00]Hi everyone.

[00:00:03]How do you simplify what we have here without the use of calculator?

[00:00:11]No calculator is allowed.

[00:00:14]So, we have um square root of 21 to power 7 minus 21 to power 6.

[00:00:25]So, what do we do?

[00:00:27]We know that this can be written as 21 to the power of 6 plus 1 because we already have 6 here, right?

[00:00:38]So, this is minus 21 to the power of 6.

[00:00:44]And um from here we apply this law of indices.

[00:00:50]A to the power of B plus C being equal to A to the power of B times A to the power of C.

[00:01:00]Right? So, if we apply that to what we have then we will be having um 21 to power 6 multiplied by 21 to the power of 1.

[00:01:12]Then we have minus 21 to the power of 6.

[00:01:18]Now, within this um under this square root sign we have um how do I call it?

[00:01:26]We have common factor 21 to the power of 6. So, we take a step so that we can have 21 to the power of 6 as the common factor.

[00:01:40]So, we bring it out. In here, we have 21 to the power of 1 minus 21 to the power of 6 divided by itself is 1.

[00:01:50]So, if we go on from here we are going to have the square root of 21 to power 6 multiplied by 21 minus 1 because 21 to the power of 1 is 21.

[00:02:08]So that from here we have 21 to the power of 6 multiplied by 20.

[00:02:16]Right?

[00:02:17]Let's continue from here.

[00:02:20]Okay, so from here what do we do?

[00:02:25]We can even split what we have here so that we have the square root of 21 to the power of 6 multiplied by the square root of 20.

[00:02:39]Okay, this is because if you have um the square root of AB you can split it and get the square root of A times the square root of B.

[00:02:51]So this is what just happened over there.

[00:02:56]Okay, so from here now we know that this 21 to the power of 6 is the same thing as 21 to the power of 3 times 2.

[00:03:10]Then multiply by 20 is 4 times 5.

[00:03:17]20 is 4 times 5.

[00:03:19]Now from here the square root here, remember that square root of A is the same thing as um A to the power of 1 over 2, right?

[00:03:30]So in the same way the square root of this will be 21 to the power of 3 times 2 raised to the power of 1 over 2.

[00:03:41]Then multiply by here we split it like we did before square root of 4 times the square root of what?

[00:03:49]5.

[00:03:51]And now [snorts] to continue from here, what do we do?

[00:03:57]We are going to multiply these two, so that this can go with this one here.

[00:04:02]So, if that happens, we have 21 to the power of 3 multiplied by the square root of 4, which is 2.

[00:04:11]Square root of 4 is 2, then multiply by the square root of 5.

[00:04:16]Now, square root of 5 is not a perfect square.

[00:04:20]Okay, 5 is not a perfect square. It's going to give us decimal figure. So, what do we do?

[00:04:27]We are going to simplify this. This can be written as 2 21 to the power of 3.

[00:04:34]Okay, the 3 now, let me break it and get um 2 + 1.

[00:04:39]Then, multiply this by 2, multiply by root 5.

[00:04:44]So, what do we do from here?

[00:04:47]Using one of the laws of indices, you know, the popular law. Think I've said this in this video already, that m to the power of a + b is m to the power of a * m to the power of b. Same thing happens here.

[00:05:03]21 to the power of 2 multiplied by 21 to the power of 1 multiplied by 2 multiplied by root 5.

[00:05:13]Now, remember we are not use calculator.

[00:05:16]So, that is the more reason we want to break everything down until we get the you know, the least value that we can get all this most simplified value.

[00:05:28]So, from here, we have our 21 squared.

[00:05:31]21 [snorts] squared, that will be 21 * 21.

[00:05:35]Then, we have this one again, which is * 21. Then, we have 2 * 2.

[00:05:43]Okay, so this is um multiplying the square root of 5 as well.

[00:05:49]And from here, 21 squared Remember that 21 squared is the same as 21 * 21, and that will give us 441.

[00:05:59]So, from here, we have 441 * this 21, then * 2.

[00:06:07]Then we multiply by root 5.

[00:06:10]So, what do we do?

[00:06:12]441 * 2. Okay, before then, let's multiply this. 21 * 2 is 40 is 42. So, we're going to have um Okay. Let me do this here.

[00:06:27]We're going to have four 41 multiplied by 42.

[00:06:33]Let's see what this will give us.

[00:06:35]Remember I'm not to use calculator, so I have to break it down. 4 * 1 is 2.

[00:06:41]Sorry, 2 * 1 is 2. 2 * 4 is 8. 2 * 4 again is 8. And we're going down to the four. 4 * 1 is 4. 4 * 4 is 16. 4 * 4 again is 16 + 1, and we have 17. Now, we add both of them.

[00:07:03]So, the addition will give us two.

[00:07:07]This is multiplication. The addition will give us two. To give 12.

[00:07:11]It will give 14, which will be 15.

[00:07:15]Then this will turn to eight, and this is one. So, what are we having? We have 100 Okay, this is 18,522.

[00:07:25]So, let me remove this.

[00:07:28]Remember it was 18,522.

[00:07:32]So, when you multiply these three, 1 2 3, you're going to get 18,522, then we have root 5.

[00:07:44]Now, we are not to use calculator to simplify this. So, if you want, you can stop at this level, or you can have everything under a single square root sign.

[00:07:56]If you stop here, this is correct.

[00:07:58]Can have this as your answer, or you express everything under the single square root sign. So, that means you're going to have the square of this.

[00:08:10]18 5 2 2, this will be squared, then you multiply it by 5.

[00:08:19]Multiply by 5. But, it's going to be a large number, so you can stop here.

[00:08:24]This is still your answer. This and this are the same.

[00:08:28]Thank you for watching. I believe you enjoyed the steps I took, all the steps I, you know, took before getting to this point. Thank you for watching. Subscribe for more.