Solve y=x^k

$$y = x ^ { k }$$

Solve for k (complex solution)

$$\left\{\begin{matrix}k=\frac{2\pi n_{1}i}{\ln(x)}+\log_{x}\left(y\right)\text{, }n_{1}\in \mathrm{Z}\text{, }&y\neq 0\text{ and }x\neq 1\text{ and }x\neq 0\\k\in \mathrm{C}\text{, }&\left(x=0\text{ and }y=0\right)\text{ or }\left(x=1\text{ and }y=1\right)\end{matrix}\right.$$

Solve for x (complex solution)

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

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

Solve for k

$$\left\{\begin{matrix}k=\log_{x}\left(y\right)\text{, }&y>0\text{ and }x\neq 1\text{ and }x>0\\k\in \mathrm{R}\text{, }&\left(x=1\text{ and }y=1\right)\text{ or }\left(x=-1\text{ and }y=-1\text{ and }Denominator(k)\text{bmod}2=1\text{ and }Numerator(k)\text{bmod}2=1\right)\\k>0\text{, }&x=0\text{ and }y=0\end{matrix}\right.$$

Solve for x

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