Olympiad Mathematics Trick | No use of calculator.

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[00:00:00]Okay, so what do you think this is equal to?

[00:00:06]Mind you, we are not to use calculator here.

[00:00:10]We're going to see how we can simplify what we have.

[00:00:16]This is 1 over 2 the square root of 3 minus 3.

[00:00:23]This is raised to the power of 6.

[00:00:27]So, how do we simplify this?

[00:00:31]First of all, we're going to deal with what we have in the bracket.

[00:00:36]Remember what we call the conjugate, right?

[00:00:40]So, we have 1 over 2 square root of 3 minus 3 then multiply this by the conjugate, which is the this denominator, but here we're going to have um addition.

[00:00:57]So, we have 2 square root of 3 plus 3 over 2 square root of 3 plus 3.

[00:01:08]So, whatever we have here is still raised to the power of um 6.

[00:01:17]Now, we're going to multiply so that the denom- the numerator will be 1 times that, that will be the same thing, 2 root 3 plus 3.

[00:01:30]This is all over the denominator we have um 2 root 3 minus 3 to multiply 2 root 3 plus 3.

[00:01:46]So, the whole of this is still raised to the power of 6.

[00:01:53]And what can you see from the denominator here? We have difference of two squares.

[00:01:59]Because a minus b multiplied by a plus b is the same thing as a squared minus b squared.

[00:02:12]And that's what we have here. So, our a now is 2 root 3 and our b is 3.

[00:02:19]So, let's express this in this form.

[00:02:22]Okay, let me remove this.

[00:02:27]So, that we can have 2 root 3 plus 3 right?

[00:02:34]Divided by here, we have 2 root 3 to the power of 2 then minus 3 to the power of 2.

[00:02:45]Remember that this is still to the power of 6.

[00:02:50]Interesting, right?

[00:02:53]So, we take a step forward and we get to square root of 3 plus 3.

[00:03:01]This is all over here. We have um You know, this is 2 to the power of 2, right? So, we have 4 multiplied by this can take this out now.

[00:03:13]So, we have 3. Then we have minus 3 squared, which is 9.

[00:03:18]Everything here is still raised to the power of 6.

[00:03:22]You know, I've not really touched the the power, right?

[00:03:27]So, from here we have 2 root 3 plus 3 over here, we have 4 times 3, that is 12.

[00:03:37]12 minus 3, 12 minus 9 is 3.

[00:03:41]So, this is now raised to the power of 6.

[00:03:46]Okay, this is raised to the power of 6.

[00:03:49]Now, see what we can do from here?

[00:03:55]Okay, so from here now You know we can write this as two root three over three plus three over three right?

[00:04:06]Yes, this is the same thing.

[00:04:09]And three into three is one.

[00:04:11]Meaning that we have two root three over three plus one is raised to the power of two.

[00:04:21]Of six rather.

[00:04:24]But we can even break the power down.

[00:04:26]Let's break the power down.

[00:04:28]So that we have two right?

[00:04:32]root three over two is over three then plus one This is raised to the power of two plus two plus two.

[00:04:46]This will give us six.

[00:04:48]And from one of the laws of indices you know that a to the power m plus n is the same thing as a to the power m times a to the power of n.

[00:05:02]I hope you remember that right?

[00:05:06]Okay, so if you do we're going to write this as two over three root three plus one This is raised to the power of two.

[00:05:20]Then multiply by the same thing two over three root three plus one raised to the second two. Why is it multiplication between the bases?

[00:05:33]Because it's addition.

[00:05:36]You know, among the powers right?

[00:05:38]So here we have two over three root three plus one and this is still raised to the power of two.

[00:05:50]Okay.

[00:05:52]Sorry, I wrote that out of sight, right?

[00:05:56]Okay. So, what we're going to do now is to expand one of this. Let's expand one of this.

[00:06:03]So, we know that a plus b to the power of two is same thing as a squared plus two a b and plus b squared, if you can remember.

[00:06:18]So, the first term here now is going to be a squared, which will be two over three root three. That's our a.

[00:06:29]Then this will be squared.

[00:06:32]Then we have plus two times a. A is this.

[00:06:37]two root three over two then multiply by b. Our b is what?

[00:06:45]B is one, so we multiply this by one.

[00:06:49]Then plus here we have b squared, which will be one squared.

[00:06:53]So, whatever we have from here now is what should be is what this represents. Like the whole of this squared. That's what this represents.

[00:07:05]Now, let's simplify this.

[00:07:08]Okay, so here we have two um two squares, that is four over three squared, that is nine.

[00:07:20]Then this can cancel this, so we multiply this by three.

[00:07:25]Then we have plus we go over to this.

[00:07:30]Oh, sorry. No, this is three.

[00:07:33]This is three here, right?

[00:07:35]So, if we open this, we have two times two, that is four over over three, then we have um root three.

[00:07:46]And this is plus one.

[00:07:49]Now, to still go on from here, what do we do? This is 12 over nine plus here we have four over three root three then we have plus one.

[00:08:04]By the way, 12 over nine, we can reduce it.

[00:08:09]We can reduce it and we get four over three.

[00:08:14]Okay, three into itself one, three into nine is three.

[00:08:18]So, this is what we have.

[00:08:21]Then, we find the LCM.

[00:08:23]The LCM of all of this is three.

[00:08:25]So, that means we have three here.

[00:08:28]And we do this.

[00:08:30]Three divided by three is one, so here we're going to have four.

[00:08:34]Then, three divided by this three is one, then here we have four root three.

[00:08:39]Then, plus three divided by Okay, just multiply three by one and that's is three.

[00:08:48]So, from here from here, see what we're going to have.

[00:08:53]We can add these and these, so we get seven.

[00:08:57]Then, we have plus four root three. Okay, I hope we can see. You know what?

[00:09:05]Let me write this better.

[00:09:08]Okay, so this plus this is going to be seven and we have plus plus four root three.

[00:09:17]This is all over three.

[00:09:19]Okay, so this is what we have in the first bracket. And we're having this in two places, in three places.

[00:09:27]Yes, we're having this in three places. So, this means that whatever we have now can be raised to the power of three.

[00:09:36]Okay, you can open the bracket and you have 7 + 4 root 3 to the power of 3 Okay, and everything will be over 3 to the power of 3, which is 27.

[00:09:52]Right? So, what are we saying now?

[00:09:55]We are saying that 1 over 2 root 3 minus 3 to the power of 6 is the same as 7 + 4 root 3 to the power of 3 all over 27. So, the two of them are the same thing from what we have been able to simplify.

[00:10:26]Now, if you want to go on with this, we can still simplify this.

[00:10:30]But, for the sake of time, we stop here.