Solve y=1/x^a

$$y = \frac { 1 } { x ^ { a } }$$

Solve for a (complex solution)

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

Solve for x (complex solution)

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

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

$$y\neq 0$$

Solve for a

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

Solve for x

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