Remix of a familiar radical equation

Part I. Tetration

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Courtesy of Mendrin's problem , I would like to consider the famous tetration/exponent tower: Notice that $\sqrt{2} = \sqrt[4]{4}$ since $\sqrt[4]{4} = (2)^{2/4} = 2^{1/2} = \sqrt{2}$ So the tetration of $\sqrt[4]{4}$ is also $2$ . The answer is $x + y = \boxed{6}$ .

Note: The reason why tetration of $\sqrt[4]{4}$ is not $4$ is due to the bounds of convergence of tetration of $x$ .

Note: There is no strong reason to consider other positive integers as shown in the next part.

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Part II. $x^y = y^x$

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Consider the classic equation $x^y = y^x$ . Condition $x$ and $y$ to be distinct positive integers. Since there are two unique integral solutions $(2,4)$ and $(4,2)$ (both of them in terms of $(x,y)$ ), we see that $2^4 = 4^2$ . If suppose we take the $\sqrt[8]{}$ from both sides, we obtain $\sqrt[8]{2^4} = \sqrt[8]{4^2} \quad \Longrightarrow \quad \sqrt{2} = \sqrt[4]{4}$

Note: You can read more about deriving the solutions here .

As a bonus, it is interesting to derive the integral solutions, using Lambert-W function . :)