Solve y=x^3m-9

$$y = x ^ { 3 m - 9 }$$

Solve for m (complex solution)

$$\left\{\begin{matrix}m=\frac{2\pi n_{1}i}{3\ln(x)}+\frac{\log_{x}\left(y\right)}{3}+3\text{, }n_{1}\in \mathrm{Z}\text{, }&y\neq 0\text{ and }x\neq 1\text{ and }x\neq 0\\m\in \mathrm{C}\text{, }&\left(x=0\text{ and }y=0\right)\text{ or }\left(x=1\text{ and }y=1\right)\end{matrix}\right.$$

Solve for x (complex solution)

$$x=e^{\frac{Im(m)arg(y)+iRe(m)arg(y)-3iarg(y)}{3\left(\left(Re(m)\right)^{2}+\left(Im(m)\right)^{2}-6Re(m)+9\right)}-\frac{2\pi n_{1}iRe(m)}{3\left(\left(Re(m)\right)^{2}+\left(Im(m)\right)^{2}-6Re(m)+9\right)}-\frac{2\pi n_{1}Im(m)}{3\left(\left(Re(m)\right)^{2}+\left(Im(m)\right)^{2}-6Re(m)+9\right)}+\frac{2\pi n_{1}i}{\left(Re(m)\right)^{2}+\left(Im(m)\right)^{2}-6Re(m)+9}}\left(|y|\right)^{\frac{Re(m)-iIm(m)-3}{3\left(\left(Re(m)\right)^{2}+\left(Im(m)\right)^{2}-6Re(m)+9\right)}}$$

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

Solve for m

$$\left\{\begin{matrix}m=\frac{\log_{x}\left(y\right)+9}{3}\text{, }&y>0\text{ and }x\neq 1\text{ and }x>0\\m\in \mathrm{R}\text{, }&\left(x=-1\text{ and }y=-1\text{ and }Numerator(3m-9)\text{bmod}2=1\text{ and }Denominator(3m)\text{bmod}2=1\right)\text{ or }\left(y=1\text{ and }x=1\right)\\m>3\text{, }&x=0\text{ and }y=0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=y^{\frac{1}{3\left(m-3\right)}}\text{, }&\left(y>0\text{ and }m\neq 3\right)\text{ or }\left(y=0\text{ and }m>3\right)\text{ or }\left(m\neq 3\text{ and }Numerator(3m-9)\text{bmod}2=1\text{ and }Denominator(3m)\text{bmod}2=1\text{ and }y<0\text{ and }Denominator(\frac{1}{3\left(m-3\right)})\text{bmod}2\neq 1\right)\text{ or }\left(Numerator(3m-9)\text{bmod}2=1\text{ and }m\neq 3\text{ and }Denominator(3m)\text{bmod}2=1\text{ and }y^{\frac{1}{3\left(m-3\right)}}\neq 0\text{ and }y<0\right)\\x=-y^{\frac{1}{3\left(m-3\right)}}\text{, }&\left(y<0\text{ and }Denominator(\frac{1}{3\left(m-3\right)})\text{bmod}2=1\text{ and }m\neq 3\text{ and }Numerator(3m-9)\text{bmod}2=1\text{ and }Numerator(3m-9)\text{bmod}2=0\text{ and }Denominator(3m)\text{bmod}2=1\text{ and }y^{\frac{1}{3\left(m-3\right)}}\neq 0\right)\text{ or }\left(m\neq 3\text{ and }y>0\text{ and }Numerator(3m-9)\text{bmod}2=0\text{ and }Denominator(3m)\text{bmod}2=1\right)\text{ or }\left(Numerator(3m-9)\text{bmod}2=0\text{ and }y=0\text{ and }m>3\right)\text{ or }\left(y>0\text{ and }m\neq 3\text{ and }y^{\frac{1}{3\left(m-3\right)}}<0\text{ and }Numerator(3m-9)\text{bmod}2=0\right)\\x\neq 0\text{, }&m=3\text{ and }y=1\end{matrix}\right.$$