Olympiad Mathematics Trick | Surdic expression | Can you?

[00:00:00]Hi everyone.

[00:00:03]Here we have a serious one. We have to evaluate what we have here.

[00:00:08]So, what do we do?

[00:00:10]Solution.

[00:00:13]Okay, um I'll just go straight to the point, right?

[00:00:18]So, from the first term here, we are going to have 1 over Okay, 1 over 2 root 2 multiplied by root 2 over root 2.

[00:00:38]So, what I'm doing is rationalization.

[00:00:42]Right? To rationalize it, you're going to pick this this sort and it multiplies both the numerator and the denominator.

[00:00:52]The power of 3 will still come down.

[00:00:55]Plus, we do the same thing there. We have 1 over the square root of 3 and it multiplies root 3 over the square root of 3 and the [clears throat] cube will still come down.

[00:01:13]This is interesting, right? So, if we multiply this, we have root 2 over 2 multiplied by root 2 * root 2 will give us 2.

[00:01:26]So, this is what we have in the first um bracket.

[00:01:32]Right? This 2 is this one, then root 2 * root 2 is 2.

[00:01:38]Then we go over to the other one. 1 multiplied by root 2, just like 1 * root 2 give us this root 2, right?

[00:01:47]>> [snorts] >> So, 1 * root 3 will give us root 3 and it's over root 3 * root 3 will give us three.

[00:01:57]So, this still has the power of three.

[00:02:03]So, at the end of from here now, we are going to have square root of two over four to the power of three.

[00:02:13]Then, we have plus the square root of three over three to the power of three.

[00:02:19]So, the next question is how then do we simplify this?

[00:02:24]And I have an idea. I would like us to simplify this using the addition of two cubes.

[00:02:31]Yes, addition of two cubes. You heard me. Remember that a cube plus b cube is what we call the addition of two cubes.

[00:02:43]And this identity is equal to a plus b multiplied by a squared plus ab plus b squared.

[00:02:55]And using what we have already, our a is root two over four.

[00:03:02]Our b is root three over three.

[00:03:07]Yes, that is what we have at this point.

[00:03:11]And what do we do?

[00:03:13]We will just put them into this identity.

[00:03:17]So, if you are ready with me, let's go.

[00:03:20]a plus b becomes root two over over four plus root three over three.

[00:03:32]This will be in the first bracket for this one. Then, I'm going over to that.

[00:03:37]a squared that's going to be root two over four.

[00:03:42]This will be squared because of the square on the a.

[00:03:46]Then, we have Oh, this identity is not it. We are supposed to have negative here, right?

[00:03:53]That is for the addition of two cubes.

[00:03:56]So, this is equally negative. So, our A is square root of two over four.

[00:04:04]And we multiply it by B, which is square root of three over three.

[00:04:10]Then we have plus B squared. What is our B?

[00:04:16]B is square root of three over three.

[00:04:20]And there's a square on it.

[00:04:22]So, you square it, just like that.

[00:04:26]You square it, and then we're going to close this.

[00:04:31]Right? What do we do?

[00:04:33]From here, you know you can find the LCM. What is the LCM?

[00:04:37]The LCM of four and three is um 12.

[00:04:41]So, you do this.

[00:04:43]12 / 4 is 3. 3 * √2, that'll be 3 √2.

[00:04:50]Then plus 12 / 3 is 4. 4 * √3, that'll be 4 √3. So, this is in the first bracket.

[00:05:02]Then in the second, we square this.

[00:05:06]√2 squared is two.

[00:05:09]Then 4 squared is 16.

[00:05:13]Minus we multiply these two. √2 * √3, that'll be √6.

[00:05:21]And then it's over 4 * 3, which is 12.

[00:05:25]Then we have plus √3 squared is also three.

[00:05:32]And 3 squared is nine. So, we close this.

[00:05:38]Okay? So, this is interesting, right?

[00:05:40]Now, what do I do from here?

[00:05:44]We're going to reduce Let's reduce this and this. So, we have three root two plus four root three divided by 12 in the first.

[00:05:58]Then in the second, we have two into two is one, two into 16 is eight.

[00:06:04]Then minus we have root six divided by 12.

[00:06:10]Then plus three into itself is one, three into nine is three. Then we close this.

[00:06:21]Let's continue.

[00:06:25]Okay, so um we're going to simplify this as well.

[00:06:30]So, let's write three root two plus four root three divided by 12.

[00:06:39]Then to simplify that we have to get our LCM, right?

[00:06:47]The LCM of eight 12 and three is 24.

[00:06:54]Right? Then we do this.

[00:06:56]24 divided by eight is three. Three times one is three.

[00:07:02]Minus 24 divided by 12 is two. Two times root six, that'll be two root six.

[00:07:09]Then plus 24 divided by three is eight. Eight times one is eight.

[00:07:16]Then we close this.

[00:07:20]From here, we still have this three root two plus four root three over 12.

[00:07:31]Okay.

[00:07:34]Okay, so we have this.

[00:07:37]Then from here, three plus eight that'll give us 11.

[00:07:43]Then we have minus two root six.

[00:07:46]All of this is over 24.

[00:07:49]And we close it.

[00:07:52]So, guess what we'll do from here.

[00:07:55]At this point, we have to multiply. So, let's multiply very quickly.

[00:08:01]So, we have three root two multiplied by 11. That will give us 33 root two.

[00:08:10]Then, three root two multiplied by um two root six. Three times two, that is six. Then, root two times root six, that will be root 12.

[00:08:25]I'm going over to four root three.

[00:08:28]Four root three multiplied by 11, that will be 44 root three.

[00:08:35]Right? Please take note of that.

[00:08:38]We have 44 root three. Then, 44 root three multiplied by Okay, four root three multiplied by two root six.

[00:08:48]That will give us negative. Four times two is eight.

[00:08:53]Then, root three times root six, that will be root 18.

[00:09:00]Okay, so we have root 18.

[00:09:03]And all of this will be over 12 multiplied by 24, and that is 288.

[00:09:15]Okay, this is interesting, right?

[00:09:18]So, what should we do now?

[00:09:20]You know, we can break this root 12, and we can break root 18.

[00:09:24]So, let's break it. We have three root two minus six root In place of root 12, I'll put four times three.

[00:09:34]Then, plus 44 We have root three minus eight root In place of root 18, I'll put 9 * 2.

[00:09:46]And all of this will be over 288.

[00:09:52]So, the next thing I will do is to do this. We have 3 root 2 minus Now, I would like to jump a step so that the video will not be too long.

[00:10:05]Now, look at this. Square root of 4 is 2. So, 2 * 6, that will give 12.

[00:10:12]Then we have root three.

[00:10:15]Then plus, we have 44.

[00:10:18]We have root three minus this. Square root of 9 is 3. 3 * 8, that is 24.

[00:10:27]Then we multiply by root two.

[00:10:30]And then all of this will be over 288.

[00:10:36]Now, what do you think we can do?

[00:10:39]We have this and this, right?

[00:10:44]Okay, so we have this and this.

[00:10:51]We have this and this to bring together, right?

[00:10:54]Oh, this is 33. So, here is 33, not just three.

[00:11:00]33. So, we have this and this to bring together, right?

[00:11:06]So, I will now have 33 root two minus 24 root two.

[00:11:16]These two will be together. Then these and these will be together, but let me write this positive first. So, we write 44 root three.

[00:11:25]Then here we have minus 12 root three.

[00:11:31]Everything is over 288.

[00:11:37]Okay, so what do I do from here?

[00:11:40]Um this minus this will give us what?

[00:11:44]That's going to give us 9. So, we have 9 root 2.

[00:11:49]Yes, we have 9 root 2.

[00:11:51]And then this minus this is going to give us 32 and it's positive. So, we write plus 32 root 3 and this is all over Okay, by the way, this is 32.

[00:12:09]So, write 32 root 3 and everything is over 288.

[00:12:15]Now, you look at this. The left-hand side I mean from here.

[00:12:20]9 and 32 do not have any common factor.

[00:12:25]Okay, so that means this is um the best we can do.

[00:12:29]And this also means that let's finish it here.

[00:12:34]This means that 1 over 2 root 2 to the power of 3 plus 1 over root 3 to the power of 3 is equal to 9 square root of 2 plus 32 plus 32 square root of 3 all over 288.

[00:13:07]Yes, thank you for watching and um if you're new to this channel consider subscribing for so that you can get to see more of our interesting videos.

[00:13:18]Thank you. See you in the next video.