Solve the equation $x^{\mathrm{x}^{4}}=4(x>0)$.

#
Answer: $x=\sqrt{2}$.

Obviously, $x=\sqrt{2}$ is a solution to the equation. The function $y=x^{x^{4}}$ is monotonically increasing on $[1 ;+\infty)$, so there are no other solutions on this interval. On the interval $(0 ; 1)$, there are also no solutions, as the values of the function on this interval are less than 1. Therefore, $x=\sqrt{2}$ is the only solution.