Solve (8n^{2})^{2}+(m^{2}n)^2=(8m)^2+(n^2m)^2

$$( 8 n ^ { 2 } ) ^ { 2 } + ( m ^ { 2 } n ) ^ { 2 } = ( 8 m ) ^ { 2 } + ( n ^ { 2 } m ) ^ { 2 }$$

Solve for m (complex solution)

$$\left\{\begin{matrix}m=-\frac{\sqrt{\frac{2\left(-\sqrt{n^{8}-256n^{6}+128n^{4}+4096}+n^{4}+64\right)}{n^{2}}}}{2}\text{; }m=\frac{\sqrt{\frac{2\left(-\sqrt{n^{8}-256n^{6}+128n^{4}+4096}+n^{4}+64\right)}{n^{2}}}}{2}\text{; }m=\frac{\sqrt{2\left(\sqrt{\left(n^{4}+64\right)^{2}-256n^{6}}+n^{4}+64\right)}}{2n}\text{; }m=-\frac{\sqrt{2\left(\sqrt{\left(n^{4}+64\right)^{2}-256n^{6}}+n^{4}+64\right)}}{2n}\text{, }&n\neq 0\\m=0\text{, }&n=0\end{matrix}\right.$$

Solve for n (complex solution)

$$\left\{\begin{matrix}n=-\frac{\left(64-m^{2}\right)^{-\frac{1}{2}}\sqrt{-2\sqrt{m^{8}-256m^{4}+16384m^{2}}-2m^{4}}}{2}\text{; }n=\frac{\left(64-m^{2}\right)^{-\frac{1}{2}}\sqrt{-2\sqrt{m^{8}-256m^{4}+16384m^{2}}-2m^{4}}}{2}\text{; }n=\frac{\left(64-m^{2}\right)^{-\frac{1}{2}}\sqrt{2\sqrt{m^{8}-256m^{4}+16384m^{2}}-2m^{4}}}{2}\text{; }n=-\frac{\left(64-m^{2}\right)^{-\frac{1}{2}}\sqrt{2\sqrt{m^{8}-256m^{4}+16384m^{2}}-2m^{4}}}{2}\text{, }&m\neq 8\text{ and }m\neq -8\\n=-8\left(m^{4}\right)^{-\frac{1}{2}}m\text{; }n=8\left(m^{4}\right)^{-\frac{1}{2}}m\text{, }&m=8\text{ or }m=-8\end{matrix}\right.$$

Solve for m

$$\left\{\begin{matrix}m=\frac{\sqrt{2\left(-\sqrt{n^{8}-256n^{6}+128n^{4}+4096}+n^{4}+64\right)}}{2|n|}\text{; }m=-\frac{\sqrt{2\left(-\sqrt{n^{8}-256n^{6}+128n^{4}+4096}+n^{4}+64\right)}}{2|n|}\text{, }&\frac{-2\sqrt{n^{8}-256n^{6}+128n^{4}+4096}+128}{n^{2}}+2n^{2}\geq 0\text{ and }\left(n^{4}+64\right)^{2}-256n^{6}\geq 0\text{ and }n\neq 0\\m=\frac{\sqrt{2\left(\sqrt{\left(n^{4}+64\right)^{2}-256n^{6}}+n^{4}+64\right)}}{2|n|}\text{; }m=-\frac{\sqrt{2\left(\sqrt{\left(n^{4}+64\right)^{2}-256n^{6}}+n^{4}+64\right)}}{2|n|}\text{, }&\left(n^{4}+64\right)^{2}-256n^{6}\geq 0\text{ and }n\neq 0\\m=0\text{, }&n=0\end{matrix}\right.$$

Solve for n

$$\left\{\begin{matrix}n=\frac{\sqrt{-\frac{2m\left(\sqrt{m^{6}-256m^{2}+16384}+m^{3}\right)}{64-m^{2}}}}{2}\text{; }n=-\frac{\sqrt{-\frac{2m\left(\sqrt{m^{6}-256m^{2}+16384}+m^{3}\right)}{64-m^{2}}}}{2}\text{, }&m^{2}\left(m^{4}-256\right)\geq -16384\text{ and }\frac{m\left(\sqrt{m^{6}-256m^{2}+16384}+m^{3}\right)}{64-m^{2}}\leq 0\text{ and }m^{6}-256m^{2}+16384\geq 0\text{ and }|m|\neq 8\\n=\frac{\sqrt{\frac{2m\left(\sqrt{m^{6}-256m^{2}+16384}-m^{3}\right)}{64-m^{2}}}}{2}\text{; }n=-\frac{\sqrt{\frac{2m\left(\sqrt{m^{6}-256m^{2}+16384}-m^{3}\right)}{64-m^{2}}}}{2}\text{, }&m^{2}\left(m^{4}-256\right)\geq -16384\text{ and }\frac{m\left(\sqrt{m^{6}-256m^{2}+16384}-m^{3}\right)}{64-m^{2}}\geq 0\text{ and }m^{6}-256m^{2}+16384\geq 0\text{ and }|m|\neq 8\\n=\frac{8}{m}\text{; }n=-\frac{8}{m}\text{, }&|m|=8\end{matrix}\right.$$