Solve {l}{{(x^2+7)}^3=4}{y=x}{text{Solvefor}ztext{where}}{z=y}

Solve for x, y, z

x=i\sqrt{7-2^{\frac{2}{3}}}\approx 2.32649929i\text{, }y=i\sqrt{7-2^{\frac{2}{3}}}\approx 2.32649929i\text{, }z=i\sqrt{7-2^{\frac{2}{3}}}\approx 2.32649929i

x=-i\sqrt{7-2^{\frac{2}{3}}}\approx -0-2.32649929i\text{, }y=-i\sqrt{7-2^{\frac{2}{3}}}\approx -0-2.32649929i\text{, }z=-i\sqrt{7-2^{\frac{2}{3}}}\approx -0-2.32649929i

x=e^{\frac{-\arctan(\frac{\sqrt{3}\sqrt[3]{4}\left(4^{\frac{2}{3}}+196-14\sqrt[3]{4}\right)}{2748})i+\pi i}{2}}\sqrt{|\frac{\sqrt[3]{4}\left(-1+\sqrt{3}i\right)-14}{2}|}\approx 0.245270756+2.802473598i\text{, }y=e^{\frac{-\arctan(\frac{\sqrt{3}\sqrt[3]{4}\left(4^{\frac{2}{3}}+196-14\sqrt[3]{4}\right)}{2748})i+\pi i}{2}}\sqrt{|\frac{\sqrt[3]{4}\left(-1+\sqrt{3}i\right)-14}{2}|}\approx 0.245270756+2.802473598i\text{, }z=e^{\frac{-\arctan(\frac{\sqrt{3}\sqrt[3]{4}\left(4^{\frac{2}{3}}+196-14\sqrt[3]{4}\right)}{2748})i+\pi i}{2}}\sqrt{|\frac{\sqrt[3]{4}\left(-1+\sqrt{3}i\right)-14}{2}|}\approx 0.245270756+2.802473598i

x=e^{\frac{-\arctan(\frac{\sqrt{3}\sqrt[3]{4}\left(4^{\frac{2}{3}}+196-14\sqrt[3]{4}\right)}{2748})i+3\pi i}{2}}\sqrt{|\frac{\sqrt[3]{4}\left(-1+\sqrt{3}i\right)-14}{2}|}\approx -0.245270756-2.802473598i\text{, }y=e^{\frac{-\arctan(\frac{\sqrt{3}\sqrt[3]{4}\left(4^{\frac{2}{3}}+196-14\sqrt[3]{4}\right)}{2748})i+3\pi i}{2}}\sqrt{|\frac{\sqrt[3]{4}\left(-1+\sqrt{3}i\right)-14}{2}|}\approx -0.245270756-2.802473598i\text{, }z=e^{\frac{-\arctan(\frac{\sqrt{3}\sqrt[3]{4}\left(4^{\frac{2}{3}}+196-14\sqrt[3]{4}\right)}{2748})i+3\pi i}{2}}\sqrt{|\frac{\sqrt[3]{4}\left(-1+\sqrt{3}i\right)-14}{2}|}\approx -0.245270756-2.802473598i

x=e^{\frac{\left(\arctan(\frac{\sqrt{3}\sqrt[3]{4}\left(4^{\frac{2}{3}}+196-14\sqrt[3]{4}\right)}{2748})+\pi \right)i}{2}}\sqrt{|\frac{-\sqrt[3]{4}\left(1+\sqrt{3}i\right)-14}{2}|}\approx -0.245270756+2.802473598i\text{, }y=e^{\frac{\left(\arctan(\frac{\sqrt{3}\sqrt[3]{4}\left(4^{\frac{2}{3}}+196-14\sqrt[3]{4}\right)}{2748})+\pi \right)i}{2}}\sqrt{|\frac{-\sqrt[3]{4}\left(1+\sqrt{3}i\right)-14}{2}|}\approx -0.245270756+2.802473598i\text{, }z=e^{\frac{\left(\arctan(\frac{\sqrt{3}\sqrt[3]{4}\left(4^{\frac{2}{3}}+196-14\sqrt[3]{4}\right)}{2748})+\pi \right)i}{2}}\sqrt{|\frac{-\sqrt[3]{4}\left(1+\sqrt{3}i\right)-14}{2}|}\approx -0.245270756+2.802473598i

x=e^{\frac{\arctan(\frac{\sqrt{3}\sqrt[3]{4}\left(4^{\frac{2}{3}}+196-14\sqrt[3]{4}\right)}{2748})i+3\pi i}{2}}\sqrt{|\frac{-\sqrt[3]{4}\left(1+\sqrt{3}i\right)-14}{2}|}\approx 0.245270756-2.802473598i\text{, }y=e^{\frac{\arctan(\frac{\sqrt{3}\sqrt[3]{4}\left(4^{\frac{2}{3}}+196-14\sqrt[3]{4}\right)}{2748})i+3\pi i}{2}}\sqrt{|\frac{-\sqrt[3]{4}\left(1+\sqrt{3}i\right)-14}{2}|}\approx 0.245270756-2.802473598i\text{, }z=e^{\frac{\arctan(\frac{\sqrt{3}\sqrt[3]{4}\left(4^{\frac{2}{3}}+196-14\sqrt[3]{4}\right)}{2748})i+3\pi i}{2}}\sqrt{|\frac{-\sqrt[3]{4}\left(1+\sqrt{3}i\right)-14}{2}|}\approx 0.245270756-2.802473598i