Solve for x
\left\{\begin{matrix}\\x=\frac{4^{\frac{2}{3}}\left(\sqrt[3]{y\sqrt{2y^{8}+z^{2}+10z+25}+z|y|+5|y|}+\sqrt[3]{-y\sqrt{2y^{8}+z^{2}+10z+25}+z|y|+5|y|}\right)}{4\sqrt[3]{y|y|}}\text{, }&\text{unconditionally}\\x=-\frac{4^{\frac{2}{3}}\sqrt[3]{\frac{y\sqrt{2y^{8}+z^{2}+10z+25}+z|y|+5|y|}{y|y|}}}{4}\text{, }&\frac{y^{6}}{8}+\frac{\left(z+5\right)^{2}}{16y^{2}}=0\text{ and }y\neq 0\end{matrix}\right.
Solve for y
\left\{\begin{matrix}\\y=\frac{2^{\frac{2}{3}}\left(\sqrt[3]{x\sqrt{32x^{8}+81z^{2}+810z+2025}+9z|x|+45|x|}+\sqrt[3]{-x\sqrt{32x^{8}+81z^{2}+810z+2025}+9z|x|+45|x|}\right)}{6\sqrt[3]{x|x|}}\text{, }&\text{unconditionally}\\y=-\frac{2^{\frac{2}{3}}\sqrt[3]{\frac{x\sqrt{32x^{8}+81z^{2}+810z+2025}+9z|x|+45|x|}{x|x|}}}{6}\text{, }&\frac{8x^{6}}{729}+\frac{\left(z+5\right)^{2}}{36x^{2}}=0\text{ and }x\neq 0\end{matrix}\right.
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