Résoudre frac{x^{-2}x^10}{(x^2)^4}=x^n

$$\frac { x ^ { - 2 } x ^ { 10 } } { ( x ^ { 2 } ) ^ { 4 } } = x ^ { n }$$

Calculer n (solution complexe)

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

Calculer x (solution complexe)

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

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

Calculer n

$$\left\{\begin{matrix}n=0\text{, }&x\neq 0\text{ and }|x|\neq 1\\n\in \mathrm{R}\text{, }&x=1\end{matrix}\right,$$

Calculer x

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