An algebra problem by Nahom Assefa

$\large \begin{aligned} x^{x^x} & = \left(\frac 12\right)^{\sqrt 2} = \left(\frac 14\right)^{\frac 12 \times \sqrt 2} = \left(\frac 14\right)^{\frac 1 {\sqrt 2}} = \left(\frac 14\right)^{\left(\frac 1 2\right)^\frac 12} = \left(\frac 14\right)^{\left(\frac 1 4\right)^{\frac 12 \times \frac 12}} = \left(\frac 14\right)^{\left(\frac 1 4\right)^\frac 14} \\ \implies x & = \boxed{\frac 14} \end{aligned}$

$(\dfrac{1}{4})^{(\dfrac{1}{4})}=\dfrac{1}{√2}$ . Therefore $(\dfrac{1}{4})^{(\dfrac{1}{√2})}=(\dfrac{1}{2})^{√2}$ . So $x=\dfrac{1}{4}=\boxed {0.25}$