Discussion
If \(\omega\) is an imaginary cube root of unity, \((1 - \omega + \omega^{2})^{3}\) equals to:
Answer: D
For the imaginary cube roots of unity \(1 + \omega + \omega^2 = 0\), so \(1 + \omega^2 = -\omega\). Then
\[ \begin{aligned} 1 - \omega + \omega^2 = (1 + \omega^2) - \omega \\ &= -2\omega \\ (-2\omega)^3 = -8\omega^3 = -8 \end{aligned} \]since \(\omega^3 = 1\). The options 8, 8 omega and 8 omega squared have the wrong sign or an unreduced power of omega: they come from treating the cube of -2 omega as positive or forgetting that omega cubed equals 1.
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