If you raise a cube root of -2 to the third power, the resulting number will have a congruent argument. We have
-2 = 2(cos 180º + i sin 180º)
Pick a number from the 4 given options, and raise it to the third power using DeMoivre's theorem. For example,
[tex]\left(\sqrt[3]{2}\left(\cos90^\circ+i\sin90^\circ\right)\right)^3=2\left(\cos(3\cdot90^\circ)+i\sin(3\cdot90^\circ)\right)=2(\cos270^\circ+i\sin270^\circ)=-2i[/tex]
so the first option is not a cube root of -2 because the argument 270º is not congruent to 180º. Similarly, the second and fourth options are also not cube roots of -2.
On the other hand,
[tex]\left(\sqrt[3]{2}(\cos60^\circ+i\sin60^\circ)\right)^3=2(\cos180^\circ+i\sin180^\circ)[/tex]
so the third option is a cube root of -2.
