Solve y=sqrt[n]{a}

$$y = \sqrt[ n ] { a }$$

Solve for a (complex solution)

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

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

$$n\neq 0$$

Solve for n (complex solution)

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

Solve for a

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

Solve for n

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