Solve {l}{x^2n=3}{y={(3x^3n)}^2}{text{Solvefor}ztext{where}}{z=y}

$$\left. \begin{array} { l } { x ^ {2 n} = 3 }\\ { y = {(3 x ^ {3 n})} ^ {2} }\\ { \text{Solve for } z \text{ where} } \\ { z = y } \end{array} \right.$$

Solve for x, y, z (complex solution)

$$x=e^{-\frac{\pi n_{1}\left(Im(n)+iRe(n)\right)}{\left(Re(n)\right)^{2}+\left(Im(n)\right)^{2}}}\times 3^{\frac{Re(n)-iIm(n)}{2\left(\left(Re(n)\right)^{2}+\left(Im(n)\right)^{2}\right)}}$$

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

$$y=9\left(e^{-\frac{\pi n_{1}\left(Im(n)+iRe(n)\right)}{\left(Re(n)\right)^{2}+\left(Im(n)\right)^{2}}}\times 3^{\frac{Re(n)-iIm(n)}{2\left(\left(Re(n)\right)^{2}+\left(Im(n)\right)^{2}\right)}}\right)^{6n}$$

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

$$z\in \cup n_{1},9\left(e^{-\frac{\pi n_{1}\left(Im(n)+iRe(n)\right)}{\left(Re(n)\right)^{2}+\left(Im(n)\right)^{2}}}\times 3^{\frac{Re(n)-iIm(n)}{2\left(\left(Re(n)\right)^{2}+\left(Im(n)\right)^{2}\right)}}\right)^{6n}$$

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

$$\left(Re(n)\right)^{2}+\left(Im(n)\right)^{2}\neq 0$$

Solve for x, y, z

$$\left\{\begin{matrix}x=\left(-\sqrt{3}\right)^{\frac{1}{n}}\text{, }y=243\text{, }z=243\text{, }&Numerator(n)\text{bmod}2=1\text{ and }Denominator(n)\text{bmod}2=1\text{ and }\left(-\sqrt{3}\right)^{\frac{1}{n}}\neq 0\\x=-\left(-\sqrt{3}\right)^{\frac{1}{n}}\text{, }y=243\text{, }z=243\text{, }&Numerator(n)\text{bmod}2=1\text{ and }Numerator(n)\text{bmod}2=0\text{ and }Denominator(n)\text{bmod}2=1\text{ and }\left(-\sqrt{3}\right)^{\frac{1}{n}}\neq 0\\x=3^{\frac{1}{2n}}\text{, }y=243\text{, }z=243\text{, }&n\neq 0\\x=-3^{\frac{1}{2n}}\text{, }y=243\text{, }z=243\text{, }&n\neq 0\text{ and }Numerator(n)\text{bmod}2=0\text{ and }Denominator(n)\text{bmod}2=1\end{matrix}\right.$$