[tex]z = r (cos \theta + i sin \theta) \\ \\ \sqrt[3]{z} = \sqrt[3]{r}(cos \frac{\theta +360k}{3} +i sin \frac{\theta +360k}{3} ) , k = 0,1,2[/tex]
Now find r and theta for -27
[tex]-27 = 27(-1 +0i) = 27(cos 180 + i sin 180)[/tex]
r = 27 , theta = 180
[tex]\sqrt[3]{-27} = \sqrt[3]{27} (cos \frac{180 +360k}{3} +i sin \frac{180 +360k}{3} )
[/tex]
[tex]= 3(cos 60 + i sin 60) , k =0 \\ \\ = 3 (cos 180 + i sin 180) , k =1 \\ \\ =3(cos 300+i sin 300), k =2[/tex]
