გადაწყვიტეთ z^frac{y{x}}=w

$$z ^ { \frac { y } { x } } = w$$

ამოხსნა w-ისთვის (complex solution)

$$w=z^{\frac{y}{x}}$$

$$x\neq 0$$

ამოხსნა x-ისთვის (complex solution)

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

ამოხსნა w-ისთვის

$$w=z^{\frac{y}{x}}$$

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

ამოხსნა x-ისთვის

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