Question

If $\omega$ is a cube root of unity.Find $(1-\omega )(1-{\omega }^2)(1-{\omega }^4)(1-{\omega }^8)$

Answer 1

If $\omega$ is a complex cube root of unity,
then ${\omega }^3=1$ and $1+\omega +{\omega }^2=0$.
now, $z=(1-\omega )(1-{\omega }^2)(1-{\omega }^4)(1-{\omega }^8)$
$\Rightarrow z=(1-\omega )(1-{\omega }^2)(1-\omega )(1-{\omega }^8)$
$\Rightarrow z=(1-\omega {)}^2(1-{\omega }^2{)}^2$
$\Rightarrow z=(1-2\omega +{\omega }^2)(1-2{\omega }^2+{\omega }^4)$
$\Rightarrow z=(1-2\omega +{\omega }^2)(1-2{\omega }^2+\omega )$
$\Rightarrow z=(-3\omega )(-3{\omega }^2)=9{\omega }^3=9$
Ans: 9