Solve x^16+y^2=1

$$x ^ { 16 } + y ^ { 2 } = 1$$

Solve for x (complex solution)

$$x\in e^{\frac{\pi i}{8}}\sqrt[16]{1-y^{2}},\sqrt[16]{1-y^{2}},\sqrt{2}\left(\frac{1}{2}+\frac{1}{2}i\right)\sqrt[16]{1-y^{2}},e^{\frac{3i\pi }{8}}\sqrt[16]{1-y^{2}},i\sqrt[16]{1-y^{2}},e^{\frac{5i\pi }{8}}\sqrt[16]{1-y^{2}},\sqrt{2}\left(-\frac{1}{2}+\frac{1}{2}i\right)\sqrt[16]{1-y^{2}},e^{\frac{7i\pi }{8}}\sqrt[16]{1-y^{2}},-\sqrt[16]{1-y^{2}},e^{\frac{9i\pi }{8}}\sqrt[16]{1-y^{2}},\sqrt{2}\left(-\frac{1}{2}-\frac{1}{2}i\right)\sqrt[16]{1-y^{2}},e^{\frac{11i\pi }{8}}\sqrt[16]{1-y^{2}},-i\sqrt[16]{1-y^{2}},e^{\frac{13i\pi }{8}}\sqrt[16]{1-y^{2}},\sqrt{2}\left(\frac{1}{2}-\frac{1}{2}i\right)\sqrt[16]{1-y^{2}},e^{\frac{15i\pi }{8}}\sqrt[16]{1-y^{2}}$$

Solve for y (complex solution)

$$y=-\sqrt{1-x^{16}}$$

$$y=\sqrt{1-x^{16}}$$

Solve for x

$$\left\{\begin{matrix}x\in \sqrt{-\sqrt{-\sqrt{-\sqrt{1-y^{2}}}}},-\sqrt{-\sqrt{-\sqrt{-\sqrt{1-y^{2}}}}},-\sqrt[4]{-\sqrt{-\sqrt{1-y^{2}}}},\sqrt[4]{-\sqrt{-\sqrt{1-y^{2}}}},-\sqrt{-\sqrt[4]{-\sqrt{1-y^{2}}}},\sqrt{-\sqrt[4]{-\sqrt{1-y^{2}}}},-\sqrt[8]{-\sqrt{1-y^{2}}},\sqrt[8]{-\sqrt{1-y^{2}}},-\sqrt{-\sqrt{-\sqrt[4]{1-y^{2}}}},\sqrt{-\sqrt{-\sqrt[4]{1-y^{2}}}},-\sqrt[4]{-\sqrt[4]{1-y^{2}}},\sqrt[4]{-\sqrt[4]{1-y^{2}}},-\sqrt{-\sqrt[8]{1-y^{2}}},\sqrt{-\sqrt[8]{1-y^{2}}}\text{, }&|y|=1\\x=\sqrt[16]{1-y^{2}}\text{; }x=-\sqrt[16]{1-y^{2}}\text{, }&|y|\leq 1\end{matrix}\right.$$

Solve for y

$$y=\sqrt{1-x^{16}}$$

$$y=-\sqrt{1-x^{16}}\text{, }|x|\leq 1$$