TITLE: How can I solve the given equation?
QUESTION [6 upvotes]: $$x^{x^{20}}=2^{\dfrac{1}{\sqrt2}}$$

Can anyone help me understand, how to solve the given equation for $x$?

REPLY [2 votes]: I do not recall how to find this answer, but the answer (over the reals) is
$$x = 2^{\tfrac{1}{8}} = \sqrt[8]{2}$$
It is clear there is only one answer over the reals because $1^{1^{20}} = 1 < 2^{1/\sqrt{2}}$, $2^{2^{20}}>2^{1/\sqrt{2}}$, and $x^{x^{20}}$ is strictly increasing over the positive reals.