Solve u^1/3+omega^2v^1/3=(-1/2+ifrac{sqrt{3}}{2})2+(-1/2-ifrac{sqrt{3}}{2})3

$$u ^ { 1 / 3 } + \omega ^ { 2 } v ^ { 1 / 3 } = ( - \frac { 1 } { 2 } + i \frac { \sqrt { 3 } } { 2 } ) 2 + ( - \frac { 1 } { 2 } - i \frac { \sqrt { 3 } } { 2 } ) 3$$

Solve for u

$$\left\{\begin{matrix}u=-\frac{\left(2\sqrt[3]{v}\omega ^{2}+5+\sqrt{3}i\right)^{3}}{8}\text{, }&arg(-\left(\sqrt[3]{v}\omega ^{2}+\frac{\sqrt{3}i}{2}+\frac{5}{2}\right))<\frac{2\pi }{3}\\u=0\text{, }&v=-\frac{10+9\sqrt{3}i}{\omega ^{6}}\text{ and }\omega \neq 0\text{ and }arg(-\frac{5+\sqrt{3}i}{2\omega ^{2}})<\frac{2\pi }{3}\end{matrix}\right.$$

Solve for v

$$v=-\frac{\left(2\sqrt[3]{u}+5+\sqrt{3}i\right)^{3}}{8\omega ^{6}}$$

$$arg(-\frac{\sqrt[3]{u}+\frac{\sqrt{3}i}{2}+\frac{5}{2}}{\omega ^{2}})<\frac{2\pi }{3}\text{ and }\omega \neq 0$$