Solve u=1/x^p

$$u = \frac { 1 } { x ^ { p } }$$

Solve for p (complex solution)

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

Solve for u (complex solution)

$$u=\frac{1}{x^{p}}$$

$$p=0\text{ or }x\neq 0$$

Solve for p

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

Solve for u

$$u=\frac{1}{x^{p}}$$

$$x>0\text{ or }\left(Denominator(p)\text{bmod}2=1\text{ and }x<0\right)$$