9000034308

Two solutions of the equation $$x^3+1+\mathrm{i}=0$$ are $$\begin{array}{rlr}{x}_1& =\sqrt[6]{2}(\cos \frac{5}{12}\pi +\mathrm{i}\sin \frac{5}{12}\pi ),& \\ x_2& =\sqrt[6]{2}(\cos \frac{13}{12}\pi +\mathrm{i}\sin \frac{13}{12}\pi ).\end{array}$$ Find the third solution.