Solve x^n=(-1/2)^n

$$x ^ { n } = ( - \frac { 1 } { 2 } ) ^ { n }$$

Solve for x (complex solution)

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

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

Solve for n

$$\left\{\begin{matrix}n=0\text{, }&x\neq 0\text{ and }|x|\neq \frac{1}{2}\\n\in \mathrm{R}\text{, }&x=-\frac{1}{2}\text{ and }Denominator(n)\text{bmod}2=1\end{matrix}\right.$$

Solve for x

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