Solve x=sqrt[x]{y}

$$x= \sqrt[ x ]{ y }$$

Solve for y (complex solution)

$$y=\left(|x|\right)^{\left(Re(\frac{1}{x})-iIm(\frac{1}{x})\right)\left(\left(Re(x)\right)^{2}+\left(Im(x)\right)^{2}\right)}e^{-2\pi Re(\frac{1}{x})n_{1}i\left(\left(Re(x)\right)^{2}+\left(Im(x)\right)^{2}\right)-2\pi Im(\frac{1}{x})n_{1}\left(\left(Re(x)\right)^{2}+\left(Im(x)\right)^{2}\right)+\left(Im(\frac{1}{x})+iRe(\frac{1}{x})\right)arg(x)\left(\left(Re(x)\right)^{2}+\left(Im(x)\right)^{2}\right)}$$

$$n_{1}\in \mathrm{Z}$$

$$x\neq 0$$

Solve for y

$$\left\{\begin{matrix}y=x^{x}\text{, }&x>0\text{ or }\left(x<0\text{ and }Denominator(x)\text{bmod}2=1\text{ and }Numerator(x)\text{bmod}2=1\right)\\y=-x^{x}\text{, }&\left(x<0\text{ and }Denominator(x)\text{bmod}2=1\text{ and }Denominator(x)\text{bmod}2=0\text{ and }Numerator(x)\text{bmod}2=1\right)\text{ or }\left(x>0\text{ and }x^{x}<0\text{ and }Denominator(x)\text{bmod}2=0\right)\text{ or }\left(x>0\text{ and }Denominator(x)\text{bmod}2=0\text{ and }Numerator(x)\text{bmod}2=1\right)\end{matrix}\right.$$