Olympiad Mathematics Trick | No use of calculator.

Added: 2026-05-28

[00:00:00]Okay, so how do you simplify this without the use of calculator?

[00:00:06]That's what we want to do.

[00:00:09]If you want to learn, then stay here.

[00:00:11]We have the square root of 4 4 4 8 8 9.

[00:00:18]This is 444,889.

[00:00:24]How do we get the square root of this without using calculator? The first thing we're going to do is there are other ways you can deal with it, but I want us to learn as we go through the steps.

[00:00:38]We can write this as 444 0 0 0 then we write plus 889.

[00:00:53]Now, let's break what we have again and we will have the square root of 4 4 4 0 0 0 plus 8 8 0 I see that I left out nine, so I'm going to add nine here.

[00:01:12]Now what we have here we've not changed the figure, so it's still the same thing.

[00:01:18]We are going to write this in standard form, so we have 4 um 4 4 times we have 10 raised to the power of three.

[00:01:31]That is 4000 then plus the same thing we have 88 times 10 and then we have plus nine. You know, this is 10 to power one, there's no point writing the power of one.

[00:01:47]Now, from here let's do this.

[00:01:50]I hope you know that we can write this um 444 like this 4 times 1 1 1 then multiply by 10 ^ 3.

[00:02:02]Then 888 is 8 * 11 * 10 then we have + 9.

[00:02:12]So, what again do we do from here?

[00:02:15]Let's look at this together.

[00:02:17]I believe we know that um 111 is the same as 999 / 9.

[00:02:27]And then 11 days is = 99 / 9.

[00:02:34]So, in place of 111, I'm going to write this and in place of 11, I'm going to write this.

[00:02:42]Now, you may not understand why I'm doing this right now.

[00:02:46]But if you stay till the end, you're going to see why I am doing this.

[00:02:51]So, we have 4 multiplied by that is 999 / 9 then * 10 ^ 3 + 8 * 99 9 / 9 * 10 and everything here is + 9.

[00:03:19]Okay, look at that.

[00:03:21]What if we find the LCM of everything?

[00:03:26]So, if we find the LCM Okay, I'm coming.

[00:03:31]If we find the LCM, we're going to have everything / 9. So, let me extend this.

[00:03:38]Everything will be over 9.

[00:03:41]And um 9 / 9 is 1 multiplied by the numerator, so we have 4 * 999 then multiply by 10 to the power of three plus the same thing we have eight times 99 times 10 then plus nine will multiply that because this is over one nine divided by one is nine times nine is 81 so we are going to proceed right we're going to proceed but we know that the square root here is for both the numerator and the denominator and nine is a perfect square so we're going to get the square root of nine and that will give us the square root of four times 99 nine times 10 to the power of three then we have um what do we do we have plus eight times 99 times 10 then we have plus 81 so all of this will now be over three the square root of nine and we can write this in another way let me write this quickly okay let's attack this in a different way um we know that 999 is the same thing as 1000 minus one which is the same thing as 10 to the power of three minus one then 99 that we have there is the same thing as 100 minus one which is the same thing as 10 to the power of two minus one so in place of that 99 I am going to write that so we have one over three square root of four right multiply by that Let me use um bracket.

[00:05:50]Write that. Which is um 10 to the power of 3 minus 1.

[00:05:56]Then we have 10 to the power of 3. So, we write it here.

[00:06:01]Then plus we have that, which is 8.

[00:06:06]Multiply by 99 is now 10 to the power of 2 minus 1.

[00:06:13]Right? So, whatever we have now is plus 81.

[00:06:19]Oh, I'm supposed to multiply by 10. So, multiply by 10.

[00:06:24]Then we write plus 81.

[00:06:28]So, what do I do?

[00:06:30]We still have 1 over 3.

[00:06:35]Then within the root, this one can multiply everything here. So, we have 4 multiply by 10 to the power of 3 * 10 to the power of 3 is 10 to the power of 6.

[00:06:47]Since we are to add the powers.

[00:06:49]Minus 1 * 10 to the power of 3 is 10 to the power of 3.

[00:06:57]So, we have this still in bracket, by the way.

[00:07:01]Okay, then we have plus 8 in bracket.

[00:07:05]This multiply by this is 10 to the power of 3.

[00:07:08]Then we have minus 10 still in bracket.

[00:07:12]Plus 81.

[00:07:14]So, what should we do from here?

[00:07:17]We are going to have 1 over 3 and we still have our square root.

[00:07:23]Now, what if we open this bracket? We'll have 4 multiply by 10 to the power of 6 minus minus 4 * 10. We still have that. 4 * 10 raised to the power of 3. Then we have plus Multiply this, we have 8 times 10 raised to power of three then minus this times this will give us 80, right?

[00:07:50]So, we write 80, then we have plus 81.

[00:07:55]So, to move on from here, this 1/3 is still coming down.

[00:08:01]Then, we have four times 10 raised to the power of six, right? Then, we have minus four times 10 raised to the power of three plus eight times 10 raised to the power of three.

[00:08:15]That will give us plus four times 10 raised to the power of what?

[00:08:20]Three.

[00:08:21]Then, we have minus 80 plus 81 is plus one.

[00:08:27]Okay.

[00:08:29]Okay, so let's keep going. We have 1/3 the square root of Um four here can be written as two squared.

[00:08:39]Then, I want to break this.

[00:08:40]10 to the power of six can be written as 10 to power three then to power two.

[00:08:48]Then, we have plus this one, I'll like to break it so that I can have two multiplied by two times 10 to the power of three.

[00:08:59]Then, we have plus one.

[00:09:01]I hope you are not um lost.

[00:09:04]This four here is two times two. This four is two squared.

[00:09:08]So, we now have 1/3 1/3 then we have From one of the laws of indices, we have the same power, so we can multiply two by 10 to power three and both of them will have the same power.

[00:09:27]Then, plus from here, what do we have? We still have the same thing, which is two times two times 10 raised to the power of 3 + I hope you know that 1 squared is the same as 1, so we can make this 1 squared because we are looking at a particular end.

[00:09:49]>> [snorts] >> Okay, so from here if we want we can you know separate this and have this 2 * 2 * 10 raised to the power of 3, right?

[00:10:03]Now from here now, let's say let's A be equal to 2 * 10 raised to the power of 3 and B to be equal to 1.

[00:10:17]Okay?

[00:10:18]Let A be that and B be equal to 1 cuz we already have 1 over there.

[00:10:23]Now, we have 1 over 3 and then our 2 * 10 to the power of 3 we said is A, so that means we're having A squared over here.

[00:10:33]Then plus here we have 2, so write the 2 then we have um the whole of this is A, right?

[00:10:42]So we write A.

[00:10:44]And we say that B is 1.

[00:10:46]If you multiply 1 by what we have here, it will be the same thing, right? So we can add B to this.

[00:10:53]>> [snorts] >> Then plus here we have 1 squared and we said that 1 is B, so we're having B squared here.

[00:11:00]Now, you discover that this is an identity in mathematics because we know that A Okay?

[00:11:10]+ B to the power of 2 is equal to A squared + 2 AB + B squared right? So this is an identity in mathematics and this means that in the root, I can write I can write this in this root.

[00:11:41]I can write it as a squared. Okay, not a squared. I can write it in this form a + b to the power of what? Two.

[00:11:53]So, we have a + b to the power of two.

[00:11:55]And this will now imply Um okay, I think I should write it down here. This will imply that we have 1 over 3.

[00:12:06]Then we have the root. What is a?

[00:12:08]We said that let a be 2 * 10 to the power 3. So, we write 2 * 10 to the power 3.

[00:12:16]Then we have plus b. Our b is 1. So, we now put square on that.

[00:12:22]Squared. Okay, so let's continue from here.

[00:12:28]Okay, so from here now we have 1 over 3.

[00:12:31]Then we have By the way, this can cancel this. So, we have 2 * 10 raised to the power of 3 + 1.

[00:12:43]So, at the end of the day we have 1 over 3 multiplied by This is 2,000 plus Okay, 2,000 plus 1.

[00:12:54]So, at the end of the day, 200 2,000 + 1 is 2,001 and we are dividing by 3.

[00:13:04]And we are dividing by 3. So, let's divide this very quickly.

[00:13:09]20 / 3 is 6. There's a remainder of 2, making this one 20.

[00:13:15]20 / 3 again is 6. There's a remainder of 2, making this one 21. 21 / 3 is 7.

[00:13:23]So, this becomes the answer.

[00:13:26]Now, if we go back If we go back to what we have done, you can see that the square root of 444,889 is equal to 667.

[00:13:43]So, you can confirm this yourself.

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