Solve z^3-1/2=(frac{sqrt{3}+i}{2})^20-operatorname{Re}(frac{sqrt{3}+i}{2})^20

$$z ^ { 3 } - \frac { 1 } { 2 } = ( \frac { \sqrt { 3 } + i } { 2 } ) ^ { 20 } - \operatorname { Re } ( \frac { \sqrt { 3 } + i } { 2 } ) ^ { 20 }$$

Solve for z

$$z=\frac{\sqrt[3]{2}\sqrt[6]{828120505233}e^{\frac{\arctan(\frac{524288\sqrt{3}}{59049})i+3\pi i}{3}}}{128}\approx -0.836191127-0.458937299i$$

$$z=\frac{4^{\frac{2}{3}}\sqrt[6]{828120505233}e^{\frac{\left(\arctan(\frac{524288\sqrt{3}}{59049})+\pi \right)i}{3}}}{256}\approx 0.020644204+0.953631408i$$

$$z=\frac{\sqrt[3]{2}\sqrt[6]{828120505233}e^{\frac{\arctan(\frac{524288\sqrt{3}}{59049})i+5\pi i}{3}}}{128}\approx 0.815546923-0.494694109i$$