<s>[INST] Solve $\left(\sqrt{\sqrt{2}-4-x}\right)+x^{\frac{1}{4}}=2^{\frac{-1}{4}}$ Solve the Equation in real/Complex numbers:
Solve $$\left(\sqrt{\sqrt{2}-4-x}\right)+x^{\frac{1}{4}}=2^{\frac{-1}{4}}$$
My try:
Letting $x=t^4$ we get
We get 
$$\left(\sqrt{\sqrt{2}-4-t^4}\right)+|t|=2^{\frac{-1}{4}}$$
Then:
$$\left(\sqrt{\sqrt{2}-4-t^4}\right)=2^{\frac{-1}{4}}-|t|$$
Squaring we get:
$$\sqrt{2}-4-t^4=t^2+\frac{1}{\sqrt{2}}-|t|2^{\frac{3}{4}}$$
$\implies$
we get
$$t^4+t^2+4=\frac{1}{\sqrt{2}}+|t|2^{\frac{3}{4}}$$
Any clue here?
 [/INST] As said in comments, I seriously wonder it the problem is correctly stated.
This equation shows two complex roots. Using Newton method, we have the awful iterates
$$\left(
\begin{array}{cc}
n & x_n \\
 0 & +1.00000+1.00000\, i \\
 1 & -3.58944+1.23135\, i \\
 2 & -0.75253+0.37882\, i \\
 3 & -1.89384+0.00982\, i \\
 4 & -1.89949+0.01512\, i \\
 5 & -1.89949+0.01510\, i
\end{array}
\right)$$
$$\left(
\begin{array}{cc}
n & x_n \\
 0 & +1.00000-1.00000\, i \\
 1 & -3.58944-1.23135\, i \\
 2 & -0.75253-0.37882\, i \\
 3 & -1.89384-0.00982\,i \\
 4 & -1.89949-0.01512\, i \\
 5 & -1.89949-0.01510\, i
\end{array}
\right)$$
</s>