Germany | Can you solve this? | Math Olympiad

Ajouté : 2026-05-07

[00:00:00]Hello, friend. You're welcome to solve this nice math problem, which is the square root of 2 - 1 ^ 6.

[00:00:08]Now, let's provide a solution from here.

[00:00:13]The first step is we can let square root of 2 - 1 be equal to X.

[00:00:22]So, the question is what is X ^ 6?

[00:00:27]Now, since we have this, let's call this equation one.

[00:00:34]Now, we have square root of 2 subtract one. This is equal to X.

[00:00:41]Now, let's take -1 on the right-hand side so that we have root two.

[00:00:48]This is equal to X + 1.

[00:00:52]This is X + 1. So, this is the same thing as X + 1.

[00:00:56]This is equal to root two.

[00:00:59]So, let's square on both sides from here.

[00:01:05]So that now X + 1 ^ 2, this is in the form of A + B raised to the power of two, so that we have A ^ 2 + B ^ 2 then + 2 AB.

[00:01:20]Let's apply this algebraic identity so that now we have X ^ 2 + 2X then + 1.

[00:01:30]This is equal to Now, let's eliminate the square root sign here. This is equal to two.

[00:01:37]So, this is to mean that we have X ^ 2.

[00:01:40]Let's take 2X + 1 on the right-hand side. So, this becomes 2 - 2X -1.

[00:01:52]So, we have X ^ 2. This is equal to 2 - 1. This is 1 then - 2 X.

[00:02:01]So, X ^ 2, this is equal to 1 - 2X.

[00:02:05]Let's call this equation two.

[00:02:10]The next step is to square on both sides.

[00:02:16]So that now this becomes we have A X ^ 2 ^ 2, this is X ^ 4.

[00:02:25]This is equal to 1 - 2X raised to the power of two.

[00:02:34]So, we have 1 - 2X ^ 2. This is in the form of A subtract B raised to the power of two, which can be expressed as A ^ 2 + B ^ 2 subtract 2 AB.

[00:02:48]Applying this algebraic identity, then we have X ^ 4.

[00:02:52]This is equal to 1 ^ 2.

[00:02:56]Then we have plus. This is 2X ^ 2.

[00:03:00]Then we have minus 2 * 1 * 2X.

[00:03:06]So, this is X ^ 4. This is equal to 1 squared, this is 1 then plus.

[00:03:13]This is 2X ^ 2. So, this becomes 4X ^ 2 then minus 2 * 2X, this is 4X.

[00:03:25]Now, this is X ^ 4. This is equal to 1 + 4X squared. This is the same thing as 2 * 2X ^ 2.

[00:03:37]-4X.

[00:03:42]But we have X squared. X squared, this is equal to 1 2X. So, let's substitute where we have X squared with 1 - 2X. So, we have X ^ 4.

[00:03:55]This is equal to 1 + 2 * 2, this is 4 into the parenthesis.

[00:04:01]This is 1 - 2X.

[00:04:04]-4X.

[00:04:07]So, this is X ^ 4. This is equal to 1 + 4 * 1, this is 4.

[00:04:14]That is + 4 * -2X. This is -8X then -4X.

[00:04:23]So, we have X ^ 4. This is equal to 1 + 4, this is 5.

[00:04:29]-8X -4X, this is -12X.

[00:04:36]Now, the question is what is X ^ 6?

[00:04:42]Now, let multiply X ^ 4 here.

[00:04:46]Multiplying by X ^ 2, this is equal to 5 -12X multiplying by X ^ 2.

[00:05:00]X ^ 4 * X ^ 2, this is in the form of A ^ N * A ^ M, which we can express as A ^ N + M.

[00:05:12]Applying this exponent property, this means that we have X ^ 4 + 2 and this is equal to 5 - 12X multiplying by X ^ 2. X ^ 2 is 1 - 2 X.

[00:05:35]So now, here we have X ^ 4 + 2. This is equal to 6 and this is equal to 5 - 12X * 1 - 2X. So, we have 5 multiplying by 1 - 2X.

[00:05:49]Then -12X into the parenthesis. This is 1 - 2X.

[00:05:57]Now, this is X ^ 6 and this is equal to 5 * 1, this is 5 -5 * -2X. This is 10X.

[00:06:08]Then -12X * 1, this is -12X then -12X * -2. This is +24 X ^ 2.

[00:06:23]So, this is X ^ 6. This is equal to Now, we have 5 -10X -12X.

[00:06:34]This is -22X.

[00:06:37]Then +24X ^ 2.

[00:06:43]The next step is that if you recall X ^ 2, this is the same thing as 1 - 2X.

[00:06:53]So, let's substitute X squared with 1 - 2X.

[00:06:57]So, this is X ^ 6. This is equal to 5 subtract 22X.

[00:07:04]Then +24 into the parenthesis. This is 1 2X.

[00:07:14]So, this is X ^ 6. This is equal to 5 subtract 22X.

[00:07:21]Then +24 * 1, this is 24.

[00:07:25]24 * -2X. This is -48X.

[00:07:34]So, we have X ^ 6. This is equal to 5 -22X -48X.

[00:07:43]This is -70X then +24.

[00:07:50]So, we have X ^ 6. This is equal to 5 + 24 and this is equal to 29 -70X.

[00:08:04]Now, again we have if you recall we had said that we let square root of 2 - 1 be equal to X.

[00:08:17]So now, here we have square root of 2 - 1 raised to the power of 6. This is equal to 29 -70 multiplying by square root of 2 1.

[00:08:34]So here, we have square root of 2 subtract one raised to the power of 6. This is equal to Now, we have 29 here.

[00:08:44]-70 * square root of 2. So, this is -70 root two.

[00:08:50]Then -70 * -1. This is +70.

[00:08:56]So, we have that square root of 2 - 1 ^ 6 This is equal to 29 + 70. This is equal to 99 -70 square root of two.

[00:09:12]And this is the solution to this square root math problem.

[00:09:19]So, kindly follow the steps.

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[00:09:29]See you in the next video.

[00:09:32]Bye-bye for now.