A Cool Exponential Equation | Algebra

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[prof.mohamad]

It is very easy my friend .
We can found exercices and we use this method but not easy like this exercice

[hairless]

it's probably easier to work just with the right side, so 1/8√2 = 2^-1/8 = (2^1/2)^-1/4 = (2^-1/2)^1/4 . well, from here we need the highest power to be equal to 4, so this is what we gonna do 1/4 = (1/√2)^4. great, now let's write it differently to remove the irrationality: (√2/2)^4. now get back to the equation: 2^-1/2 = √2^-1 = √2/2, so our final equation looks like this: x^x^4 = (√2/2)^(√2/2)^4 which means x is equal to √2/2, hooray! i think this solution is a bit easier, but i might be wrong)

[gelbkehlchen]

Solution:
x^(x^4) = 1/2^(1/8) |()^4 ⟹
(x^4)^(x^4) = (1/2)^(1/2) |the same operations are done with x^4 on the left side of the equation as with 1/2 on the right side of the equation, therefore ⟹ x^4 = 1/2 |()^(1/4) ⟹
x1/2 = ±1/2^(1/4) and x3/4 = ±1/2^(1/4)*i