An Amazing Math Olympiad Challenge | Can You Crack This?

Añadido: 2026-05-31

[00:00:02]Hello friends, welcome to NC Gyan. Today in this video we have another very interesting question from math Olympiad.

[00:00:09]Here we have X equal to a square root 2 plus cube root 3 and it is said to find this required expression. I will call required expression equal to E.

[00:00:20]Now we have X which is sum of two irrational numbers.

[00:00:27]Now I will calculate X power 6 from X.

[00:00:31]So I need to write over here X equal to a square root 2 plus cube root 3.

[00:00:41]Now I will cube both sides.

[00:00:45]whole cube whole cube Now here we'll be using A plus B whole cube identity. Let me write here so that we can apply directly to our equation.

[00:00:56]A plus B whole cube is A cube plus 3 A squared B plus 3 AB squared plus B cube.

[00:01:10]Here A is a square root 2 and B is cube root 3. We will get X cube.

[00:01:19]root 2 cube root 2 times root 2 times root 2 2 root 2 2 root 2 plus 3 times A squared root 2 squared will be 2 times B will write cube root 3 plus 3 times root 2 times B squared cube root 3 whole squared cube root 9 plus B cube cube root 3 whole cube 3.

[00:01:48]In this equation we have cube root 3 and cube root 9 which we want to replace from the known equation.

[00:01:59]So, from this equation I can write cube root 3 which will be equal to x minus square root Cube root 3 will be x minus square root Let me write here.

[00:02:17]3 equal to x minus square root Now, cube root 9 is also required. So, I need to square both sides.

[00:02:28]So, I will put power 2.

[00:02:31]Cube root 3 whole square is cube root 9.

[00:02:35]In RHS we'll be using a minus b whole square formula and get x square minus 2 root 2 x plus 2.

[00:02:46]Now, we have both the values.

[00:02:49]Cube root 3 which is here.

[00:02:53]And cube root 9 which is here.

[00:02:58]We are going to use x cube equation.

[00:03:02]Let me write here. x cube equal to 2 square root 2 plus 6 times cube root 3 plus 3 times square root 2 times cube root 9 plus 3.

[00:03:22]We have cube root 3.

[00:03:26]x minus root 2 And we have cube root 9.

[00:03:34]x square minus 2 root 2 x plus 2 Now, we have to plug in these two values to our equation and we will get x cube equal to two square root Now six times cube root three, we will write six times cube root three is x minus square root two.

[00:04:04]plus three root two times cube root nine Here we will write x is square minus two root two times x plus two.

[00:04:21]Then we have plus three also.

[00:04:25]Now we have to simplify our right hand side.

[00:04:29]We will get two root two six times x minus root two We'll write six x minus six root two.

[00:04:41]Three root two times x is square, we'll write three root two x is square.

[00:04:48]Then three root two times minus two root two x, we will get 12 x with negative sign, minus 12 x.

[00:04:58]Then we have three root two times two, which is six root two.

[00:05:05]plus three Now we have plus and minus six root two over here.

[00:05:11]Which we will cancel.

[00:05:14]Here we have six x.

[00:05:16]And here we have minus 12 x.

[00:05:20]So I can write here the last number or term that is three minus six x.

[00:05:29]Now we have two root two left and three root two x is square left. From these two terms I will take root two common out. So in the bracket I will be writing three x is square plus two.

[00:05:43]And in LHS we have x cube.

[00:05:47]Now we are going to add 6x -3 to both sides. Let's add 6x -3.

[00:05:58]Or here we will write 6x -3.

[00:06:02]So, plus and minus 6x, plus and minus 3 will get over.

[00:06:07]We will get x cube +6x -3 equal to square root of 2 3x squared +2.

[00:06:23]Now, we have only one irrational term, and once we will take a square both sides this will resolve.

[00:06:31]It will be only rational numbers.

[00:06:35]Let's square both sides.

[00:06:40]So, I will write here whole part two.

[00:06:44]Write here whole part two.

[00:06:46]Now, in LHS we'll be using a + b + c whole squared.

[00:06:51]And in RHS we'll be using a + b whole squared formula.

[00:06:56]So, let's use in LHS first x cube whole squared x power 6 + 6x whole squared we'll write 36x squared + -3 whole squared so +9 +2 times x cube times 6x we will get 12x power 4.

[00:07:19]Then 2 times 6x times -3 we'll get -36x.

[00:07:26]Then we will write -6x cube.

[00:07:30]-6x cube.

[00:07:33]Then in RHS square root 2 whole squared will be 2.

[00:07:38]In bracket 3x squared +2 whole squared will be 9x power 4 + 12 x square plus four.

[00:07:51]Now we will expand RHS and we will get 18 x power four plus 24 x square plus eight.

[00:08:06]Now we will take all the terms to LHS.

[00:08:09]We will get x power six.

[00:08:14]Then we can write x power four terms. So I will be writing here 12 x power four minus 18 x power four.

[00:08:24]x power four is over.

[00:08:26]Now we have x cube term which we will write over here minus six x cube.

[00:08:32]Then we have x term and x square term and constant.

[00:08:38]So I will write here x square term first.

[00:08:42]36 x square minus 24 x square then we have minus 36 x then we have two constants nine and minus eight.

[00:08:56]And right hand side will be zero now.

[00:09:00]Now we will write x power six first.

[00:09:04]Then we have 12 x power four minus 18 x power four which is minus six x power four and it is matching with our required expression over here. Then we have minus six x cube.

[00:09:20]Then we have 36 minus 24 12 x square.

[00:09:28]Then we have minus 36 x.

[00:09:35]Then we have nine - 8 1.

[00:09:39]It is completely matching with our required expression equal to zero.

[00:09:45]And here we can write e equal to zero.

[00:09:48]So our final answer becomes required expression will be equal to zero. I hope friends you will like this video. Thank you so very much for watching. Do not forget to like, share, subscribe. Bye-bye.