By DeMoivre's theorem,
[tex]z=2\mathrm{cis}60^\circ\implies z^4=2^4\mathrm{cis}(4\times60)^\circ=16\mathrm{cis}\,240^\circ[/tex]
[tex]z^4=16(\cos240^\circ+i\sin240^\circ)[/tex]
[tex]z^4=16\left(-\dfrac12-i\dfrac{\sqrt3}2\right)[/tex]
[tex]z^4=-8-i8\sqrt3[/tex]
