Germany | Can you solve this? | Math Olympiad

[00:00:00]Hello everyone, welcome to how to solve this very nice expression root 3 plus I divided by 2 whole power 2027.

[00:00:09]So, let's start.

[00:00:11]First, we suppose that let X is equal to root 3 plus I divided by 2.

[00:00:24]It means that now we have to find the value of X to the power 20 27.

[00:00:32]Now, we square both sides of this equation. This will become X squared equal to root 3 plus I divided by 2 whole squared.

[00:00:50]Expand this numerator root 3 plus I by using this algebraic identity A plus B whole squared equal to A squared plus 2 AB plus B squared.

[00:01:07]This will become X squared equal to root 3 squared plus 2 * root 3 * I plus I squared divided by 2 squared is uh 4.

[00:01:30]Next, X squared equal to this square will be canceled out with this square root and here we are left with 3. Plus, this will become 2 * root 3 * I.

[00:01:47]And this I squared I squared equal to -1.

[00:01:55]So, this uh plus I squared will become -1 divided by four.

[00:02:07]Further simplify, this will become x squared equal to 3 -1 will become two.

[00:02:14]Plus two times two times root three I divided by four.

[00:02:23]Next, x squared equal to From these two terms, we can factor out two. In bracket left, 1 plus root three times I divided by This four is same as two times uh two. So, this two will be canceled out with this two and we get the value of x squared equal to 1 plus root three I divided by two.

[00:02:57]Now, we take a cube of both sides of uh this equation, x is equal to root three plus I over two.

[00:03:12]We have x is equal to root three plus I over two. And we take uh cube of both sides.

[00:03:26]So, this will become x cubed equal to Expand this root three plus I whole cubed by using this algebraic identity, a plus b whole cubed equal to a cubed plus three times a squared b plus three times a b squared plus b cubed.

[00:03:53]So, this is root three plus i whole cubed will become root three whole cubed.

[00:04:02]Plus three times root three squared times i plus three times root three times i squared plus i cubed.

[00:04:20]Divided by two cubed is equal to eight.

[00:04:28]Next, x cubed equal to root three cubed will become three times root three.

[00:04:38]Plus this is squared will be cancelled out with this square root and three times three times i will become nine i.

[00:04:48]And three times root three times i squared this i squared is equal to negative one.

[00:04:58]So, three times root three times negative one will become negative three times root three.

[00:05:08]Plus this i cubed can be written as i squared times i.

[00:05:16]Divided by eight.

[00:05:20]Next, x cubed equal to this negative three times root three will be cancelled out with this positive three times root three.

[00:05:29]And in the numerator we are left with the nine times i since i squared is equal to negative one so negative one times i will become negative i.

[00:05:43]Divided by eight.

[00:05:50]Next, x cubed equal to 9i - 1i will become 8 * i / 8. So, this 8 will be canceled out and we are left with x cubed equal to i.

[00:06:11]Now, we take a square of both sides.

[00:06:18]So, x cubed whole square will become x to the power 6 equal to i squared.

[00:06:27]Since i squared equal to -1, so x to the power 6 will be equal to 1.

[00:06:36]And we have to calculate the value of x to the power 2027.

[00:06:43]Since 2027 2027 is equal to 6 * 337 + 5.

[00:06:56]So, we can rewrite this x to the power 2027 as x to the power 6 * 300 37 * x to the power 5.

[00:07:12]This is equal to x to the power 2027.

[00:07:20]And this x to the power 2027 can be written as x to the power 6 whole power 337 * x to the power 5. Now, we calculate the value of this x to the power 5.

[00:07:39]To calculate the value of x to the power 5, we multiply this equation x cubed equal to i by this equation.

[00:07:57]x squared equal to 1 + root 3 i over 2.

[00:08:08]We have two equations x cubed equal to i and x squared equal to 1 + root 3 * i divided by 2. We multiply these two equations to get x to the power 5.

[00:08:26]x to the power 5 is equal to x squared * x cubed.

[00:08:33]So, this will become 1 + root 3 i divided by 2 times i.

[00:08:45]So, 1 * i is i and root 3 * i * i will become + root 3 i squared divided by 2.

[00:09:04]And because i squared equal to -1, so this will become x to the power 5 is equal to i minus root 3 divided by 2.

[00:09:25]Now, because we have x to the power 5 equal to i minus root 3 divided by 2, so we replace this x to the power 5 with this uh i minus root 3 over 2.

[00:09:41]And we replace this x to the power 6 with this negative 1.

[00:09:48]So, this will become negative 1 to the power 337 * x to the power 5 is uh i minus root 3 divided by 2.

[00:10:10]Since this uh 337 is an odd number, so this negative 1 remains negative 1.

[00:10:17]This will become negative 1 * i minus root 3 divided by 2.

[00:10:27]And uh negative 1 * i will become positive i, and this negative root 3 will become positive root 3 minus i divided by 2.

[00:10:41]And this expression is equal to this uh x to the power 2027, x to the power 2027.

[00:10:50]Since uh we have supposed that x to the power 2027 equal to this given expression.

[00:11:05]So, the given expression root 3 plus i divided by 2 whole power 2027 is equal to root 3 minus I divided by 2.

[00:11:35]This is the final answer of this problem.