Solve x^8-9z^4+8=0

$$x ^ { 8 } - 9 z ^ { 4 } + 8 = 0$$

Solve for x (complex solution)

$$x\in \sqrt{2}\left(\frac{1}{2}+\frac{1}{2}i\right)\sqrt[8]{9z^{4}-8},\sqrt[8]{9z^{4}-8},i\sqrt[8]{9z^{4}-8},\sqrt{2}\left(-\frac{1}{2}+\frac{1}{2}i\right)\sqrt[8]{9z^{4}-8},-\sqrt[8]{9z^{4}-8},\sqrt{2}\left(-\frac{1}{2}-\frac{1}{2}i\right)\sqrt[8]{9z^{4}-8},-i\sqrt[8]{9z^{4}-8},\sqrt{2}\left(\frac{1}{2}-\frac{1}{2}i\right)\sqrt[8]{9z^{4}-8}$$

Solve for z (complex solution)

$$z=\frac{\sqrt{3}i\sqrt[4]{x^{8}+8}}{3}$$

$$z=\frac{\sqrt{3}\sqrt[4]{x^{8}+8}}{3}$$

$$z=-\frac{\sqrt{3}\sqrt[4]{x^{8}+8}}{3}$$

$$z=-\frac{\sqrt{3}i\sqrt[4]{x^{8}+8}}{3}$$

Solve for x

$$\left\{\begin{matrix}x=\sqrt{-\sqrt{-\sqrt{9z^{4}-8}}}\text{; }x=-\sqrt{-\sqrt{-\sqrt{9z^{4}-8}}}\text{; }x=-\sqrt[4]{-\sqrt{9z^{4}-8}}\text{; }x=\sqrt[4]{-\sqrt{9z^{4}-8}}\text{, }&\left(\frac{\sqrt{3}\times 2^{\frac{3}{4}}}{3}<\frac{\sqrt{3}\sqrt[4]{8}}{3}\text{ and }z=-\frac{\sqrt{3}\times 2^{\frac{3}{4}}}{3}\text{ and }-\frac{\sqrt{3}\sqrt[4]{8}}{3}=-\frac{\sqrt{3}\times 2^{\frac{3}{4}}}{3}\text{ and }-\frac{\sqrt{3}\times 2^{\frac{3}{4}}}{3}=-\frac{\sqrt{3}\sqrt[4]{8}}{3}\right)\text{ or }\left(\frac{\sqrt{3}\times 2^{\frac{3}{4}}}{3}<\frac{\sqrt{3}\sqrt[4]{8}}{3}\text{ and }z=-\frac{\sqrt{3}\times 2^{\frac{3}{4}}}{3}\text{ and }-\frac{\sqrt{3}\times 2^{\frac{3}{4}}}{3}\leq -\frac{\sqrt{3}\sqrt[4]{8}}{3}\right)\text{ or }\left(z=-\frac{\sqrt{3}\sqrt[4]{8}}{3}\text{ and }-\frac{\sqrt{3}\sqrt[4]{8}}{3}=-\frac{\sqrt{3}\times 2^{\frac{3}{4}}}{3}\right)\text{ or }\left(\frac{\sqrt{3}\sqrt[4]{8}}{3}=\frac{\sqrt{3}\times 2^{\frac{3}{4}}}{3}\text{ and }z=\frac{\sqrt{3}\sqrt[4]{8}}{3}\right)\\x=\sqrt{-\sqrt[4]{9z^{4}-8}}\text{; }x=-\sqrt{-\sqrt[4]{9z^{4}-8}}\text{, }&\left(\frac{\sqrt{3}\times 2^{\frac{3}{4}}}{3}<\frac{\sqrt{3}\sqrt[4]{8}}{3}\text{ and }z=-\frac{\sqrt{3}\times 2^{\frac{3}{4}}}{3}\text{ and }-\frac{\sqrt{3}\times 2^{\frac{3}{4}}}{3}\leq -\frac{\sqrt{3}\sqrt[4]{8}}{3}\right)\text{ or }\left(z=-\frac{\sqrt{3}\sqrt[4]{8}}{3}\text{ and }-\frac{\sqrt{3}\sqrt[4]{8}}{3}=-\frac{\sqrt{3}\times 2^{\frac{3}{4}}}{3}\right)\text{ or }\left(\frac{\sqrt{3}\sqrt[4]{8}}{3}=\frac{\sqrt{3}\times 2^{\frac{3}{4}}}{3}\text{ and }z=\frac{\sqrt{3}\sqrt[4]{8}}{3}\right)\\x=\sqrt[8]{9z^{4}-8}\text{; }x=-\sqrt[8]{9z^{4}-8}\text{, }&|z|\geq \frac{\sqrt{3}\times 2^{\frac{3}{4}}}{3}\end{matrix}\right.$$

Solve for z

$$z=\frac{\sqrt{3}\sqrt[4]{x^{8}+8}}{3}$$

$$z=-\frac{\sqrt{3}\sqrt[4]{x^{8}+8}}{3}$$