If [tex]z^3=64(\cos240^\circ+i\sin240^\circ)[/tex], then
[tex]z=64^{1/3}\bigg(\cos\dfrac{(240+360n)^\circ}3+i\sin\dfrac{(240+360n)^\circ}3\bigg)[/tex]
[tex]z=4\bigg(\cos(80+120n)^\circ+i\sin(80+120n)^\circ\bigg)[/tex]
for [tex]n=0,1,2[/tex]. We get
[tex]n=0\implies z=4(\cos80^\circ+i\sin80^\circ)[/tex]
[tex]n=1\implies z=4(\cos200^\circ+i\sin200^\circ)[/tex]
[tex]n=2\implies z=4(\cos320^\circ+i\sin320^\circ)[/tex]
