Olympiad Mathematics | No use of calculator | Can you solve this?

Añadido: 2026-05-22

[00:00:02]What do you think is the value of this without using calculator?

[00:00:07]Okay, let's work this together.

[00:00:10]But remember that this is going to involve a lot of manipulations.

[00:00:15]Okay, we have 4 4 4 4 1 2 3 4 5 then we have 5.

[00:00:26]This is 4 billion 444,000 Okay, 4 billion 444 million 222,225.

[00:00:34][snorts] So, we want to find the square root of this number.

[00:00:43]And this is what we will do first.

[00:00:47]4 4 4 4 1 2 3 4 5 6, let's make them zero.

[00:00:56]Okay, then we have plus. Look at this 1 2 3 4 5, let's write the two.

[00:01:03]1 2 3 4 5, then in place of this five put zero so [snorts] that we can have plus five.

[00:01:13]Now, you're going to understand what I'm doing at the end of the video. Yes, for those of you you know, some of you will be able to understand it at the end of the video. So, what do I do from here?

[00:01:26]Let's write each of them in standard form.

[00:01:30]So, we have 4 4 4 4 multiply by 10 to the power of six.

[00:01:39]1 2 3 4 5 6, okay?

[00:01:42]Plus there we have two two two two two multiply by 10.

[00:01:50]Then we have plus five.

[00:01:53]Now, what do we do from here?

[00:01:56]We're going to simplify this.

[00:02:01]Okay, so that we can have four multiplied by 1 2 3 4.

[00:02:06]Right? And whatever we have will be multiplied by 10 to power 6.

[00:02:11]This times this will give you 4 4 4 4.

[00:02:14]Plus, here we are going to have the same thing two into 1 2 3 4 5.

[00:02:21]1 2 3 4 5.

[00:02:24]Right? Then multiply by 10.

[00:02:27]This is all over what?

[00:02:31]Okay, we've not divided, right? So, we've not divided.

[00:02:34]Let's continue.

[00:02:36]And we have plus five.

[00:02:40]So, [snorts] what again do we do?

[00:02:42]Let's look at this 1 1 1 1 and this 1 1 1 1 1.

[00:02:48]>> [snorts] >> We can do something there.

[00:02:50]Okay, and here is what we will do.

[00:02:53]We're going to have four multiplied by This is 9 9 9 9 divided by 9.

[00:03:02]This will give you four 1 1 1 1. We're multiplying by 10 to power 6.

[00:03:08]Then plus the same thing happens here.

[00:03:11]We have two multiplied by 1 2 3 4 5.

[00:03:17]This is all over what? Nine.

[00:03:19]If you divide this, you're going to get this 11,111.

[00:03:25]Then whatever we'll multiply by 10.

[00:03:29]Then we have our plus five.

[00:03:32]Now, let's do something.

[00:03:34]Let's do something again. This can be written in this form.

[00:03:40]We have four.

[00:03:41]This 9,999 is the same as 10,000 minus one.

[00:03:49]So, this is over nine.

[00:03:52]And we see multiply by 10 to the power of six.

[00:03:56]Then we have plus two into Okay.

[00:04:02]We have plus two into This 99,999 is the same thing as 100,000 minus one.

[00:04:14]Okay, remember it's still over nine.

[00:04:16]And we multiply by 10 plus our five.

[00:04:22]Okay, this is interesting, right?

[00:04:25]Very interesting. So, that if we go on from here we shall have Okay, let me do this better.

[00:04:34]Remember, this is nine.

[00:04:38]So, from here now we're going to get four multiply by 10,000 in index form is 10 to the power of four. We have minus one.

[00:04:47]And this is over nine.

[00:04:49]And we will still multiply it by 10 to the power of six.

[00:04:55]Okay. If you like, you remove the brackets, it's the same thing. Then we have plus two multiply by Um 100,000 is 10 to the power of five.

[00:05:07]Then we have minus one.

[00:05:10]This is all over nine multiply by 10 and we have plus five.

[00:05:17]Interesting. So, at this point we will open the brackets.

[00:05:23]Yes, we are going to open the brackets, so that we will have the square root of four.

[00:05:32]Okay.

[00:05:34]Um This 10 to the power of four is going to multiply Um 10 to the power of four is going to multiply what we have here, 10 to the power of six.

[00:05:46]And that will give for us 10 to the power of 10.

[00:05:50]Pick one of the bases and add the powers.

[00:05:53]Minus 1 * 10 to the power of six is 10 to the power of six.

[00:05:57]So, close this. This is still over nine.

[00:06:00]Then we have plus two.

[00:06:03]Right? Then 10 to the power of five times 10 will give us 10 to the power of six.

[00:06:08]Minus 1 * 10 is 10.

[00:06:12]This is still over nine, and we have our plus five.

[00:06:18]Okay. So, at this point, what should we do?

[00:06:22]Let me set this very well.

[00:06:25]Okay, what do we do at this point?

[00:06:29]Okay, so from here we're going to open the brackets now.

[00:06:33]As we have 4 * 10 to the power of 10, that is four times 10 to the power of 10 minus four again times 10 to the power of six.

[00:06:45]This is all over nine.

[00:06:47]Then plus two times 10 to the power of six minus two times that is 20.

[00:06:53]This is over nine.

[00:06:55]And we have plus five.

[00:06:58]Now, we take out the LCM. The LCM is nine.

[00:07:02]So, we will now have four times 10 to the power of 10 minus four times 10 to the power of six plus two times 10 to the power of six minus 20 then plus 45.

[00:07:25]Remember, this is over one. And nine divided by nine one is nine. Nine times five is 45. So, everything here will now be over nine.

[00:07:37]Okay, so we continue.

[00:07:41]We have the square root.

[00:07:43]Now, we have our four * 10 ^ 10.

[00:07:49]Then, look at the This is four * 10 ^ 6 - 4 um 2 * 10 ^ 6. This is going to give us - 2 * 10 ^ 6.

[00:08:02]Right? And then, - 20 + 45. That will give us + 25.

[00:08:11]And all of these will be over nine.

[00:08:15]Right? All of these is over over nine. This is interesting.

[00:08:22]Okay, so from here let's do this.

[00:08:27]If you're still getting this up to this lens, then you're doing really well.

[00:08:31]Four [snorts] here can be written as two squared. Then multiply by 10.

[00:08:36]This power of 10, let's break it as 5 * 2.

[00:08:41]Okay, then we have - 2 * 10 ^ 6. But this six, we're going to break it as 5 + 1.

[00:08:50]Then we have + 25.

[00:08:55]Yes, we have + 25. So, to continue from here, what do we do?

[00:09:02]Okay, to continue from here, we're going to work on this to get um the square root of 2 ^ 2.

[00:09:10]Right? Let's separate this so that we have 10 ^ 5.

[00:09:16]And then there's a square on it, right?

[00:09:18]We have - 2 * 10 ^ 5.

[00:09:23]Remember from the law of indices this is 10 ^ 5 * 10 ^ 1.

[00:09:29]10 ^ 1 is 10, so this is multiplied by 10. And we have our + 25.

[00:09:36]Remember that all of this is still over nine.

[00:09:39]Oh, I didn't put that, you know, the last step.

[00:09:43]Remember that all of this is divided by nine.

[00:09:47]Yes.

[00:09:48]That is divided by nine. Now, let's look at this very well.

[00:09:52]Okay, from one of the laws of indices, I think we know what to do here.

[00:09:57]This is um the square root of the same powers, right? So, we can multiply the bases.

[00:10:05]Okay, and then square No, raise to the same power.

[00:10:09]Okay, we can do that. Then, on the other hand, we will still have the same thing, which is um two times 10 raised to power five, then we still multiply by 10 plus our 25.

[00:10:26]Don't forget that all of this is still over nine.

[00:10:30]Right? The nine is still under the square root sign.

[00:10:34]Don't forget.

[00:10:37]Okay, so let's take this step right away. This is the square root of from here we have two times 10 raised to power five squared.

[00:10:48]Then minus Look at this. This 10 I can break it to be two times five. So, I'll write that two first. Then this will be in bracket two times 10 to power five.

[00:11:00]Then we multiply by five now.

[00:11:03]Okay, because two times five will give that 10. This is plus 25. Why don't I write 25 as five squared?

[00:11:13]Okay, I can write 25 as five squared, right?

[00:11:16]So, from here now, what do we see?

[00:11:20]What do we see?

[00:11:22]Okay, if you look at this very well, you should understand something now. Let's say Let A be equal to two.

[00:11:31]Okay, let A be equal to two multiplied by 10 to the power of five.

[00:11:38]Okay, what we have in the bracket there.

[00:11:40]And our B and B to be equal to five.

[00:11:48]So, let A be equal to 2 * 10 ^ 5 and B to be equal to 5.

[00:11:55]Okay, so our A is this and B is this.

[00:11:59]So, this means that this is what we're going to have.

[00:12:03]We're going to have in place of this, we're going to write A squared because A is 2 * 10 ^ 5 then minus two.

[00:12:14]Okay. Now, this is Okay, let me first remove the brackets, right?

[00:12:19]2 * 10 ^ 5, we said this is A, right?

[00:12:22]And 5 is what? B.

[00:12:25]Then we have plus 5 squared is going to be B squared according to what we have said.

[00:12:32]Okay, so I believe you're trying to get this by now. And from here, remember that if you have A minus B squared this is the same thing as A A um squared minus 2 AB plus B squared. The same thing that we have here.

[00:12:54]And this means that we can now write this as in place of um the whole of this, we'll write A minus B to the power of two.

[00:13:07]Yes, this is what we're going to write.

[00:13:10]Right? Oh, I think there's um a mistake, right? Yes, there's a mistake. Let me point it out. There's a mistake. The whole of this you know, we're still dividing it by nine.

[00:13:23]Okay, we're still dividing the whole of that by nine.

[00:13:29]Okay, it is under the square root sign.

[00:13:32]Okay?

[00:13:33]And that means that whatever we have here the whole of this we are still dividing by nine.

[00:13:41]Yes.

[00:13:43]So, that means that what we have here right now will still be divided by nine. I don't know if you can see that.

[00:13:52]So, this will still be divided by nine.

[00:13:55]So, what do we do?

[00:13:59]Still under the square root sign, but our A is um 2 * 10 ^ 5. So, let's put it back into this.

[00:14:09]Okay, let me write it here.

[00:14:11]This will now be the square root of A, which is two times 10 ^ 5 minus B, which is what? Five.

[00:14:23]And this is um Okay, this is raised to the power of two, right?

[00:14:29]And it's over nine.

[00:14:31]But we can combine what we have here.

[00:14:34]So, that we can have the square root of from one of the laws of indices, we have two times 10 ^ 5 minus five over Look at nine. Nine is three squared. So, this will be combined to the power of two.

[00:14:52]So, the square root and this can go so that we can have two times 10 raised to the power of five minus five all over three. We're getting to the end.

[00:15:06]I believe you can understand this.

[00:15:09]We're getting closer to the end. So, that from here our two times 10 raised to the power of five is the same as two 000 200000 five zeros because of 10 to the power 5.

[00:15:26]Then this is um -5.

[00:15:29]And we are dividing by 3.

[00:15:32]Okay? So, if we subtract um 200,000 you know, take 5 out of 200,000, what are we going to have? If we do that, we are going to have 1 9999 5. And that is 199,995.

[00:15:56]Take one away. I mean, take We've taken five away. This is all over 3.

[00:16:03]So, let us divide what we have here.

[00:16:06]Let us divide. 19 / 3, that will give us 6.

[00:16:12]Remainder one, making this to be 19. 19 / 3 again is 6. Remainder one, making this to be 19. 19 / 3 is 6 again.

[00:16:22]Remainder one, making this to be 19. 19 / 3 is another 6.

[00:16:28]Remainder one, making this to be 15. And 15 / 3 is equal to 5. There is no remainder.

[00:16:37]So, this right now is the answer.

[00:16:41]Okay, so from our work now, the square root of this number is equal to 66665.

[00:16:50]And that is 66,665.

[00:16:53]So, you can confirm it yourself.

[00:16:55]Thank you for watching.