If $\omega$ is a complex cube root of unity, find the value of the product $(1-\omega )(1-{\omega }^2)(1-{\omega }^4)(1-{\omega }^8)$.

Solution

1. Note that ${\omega }^4=\omega$ and ${\omega }^8={\omega }^2$.
2. The product is $(1-\omega )(1-{\omega }^2)(1-\omega )(1-{\omega }^2)=[(1-\omega )(1-{\omega }^2){]}^2$.
3. Expand the inner term: $$\begin{array}{rl}1-{\omega }^2-\omega +{\omega }^3& =1-({\omega }^2+\omega )+1\\ & =1-(-1)+1\\ & =3\end{array}$$.
4. The total product is $3^2=9$.
5. Hence the answer is (B).