Question

Let $\omega$ being the cube root of unity then find the value of $(1-\omega )(1-{\omega }^2)(1+{\omega }^4)(1+{\omega }^8)$.

Answer 1

Now, $(1-\omega )(1-{\omega }^2)(1+{\omega }^9)(1+{\omega }^8)$

$=1-\omega -{\omega }^2+{\omega }^{3}1+{\omega }^4+{\omega }^8+{\omega }^{12}$

$=1-(\omega -{\omega }^2)+11+(\omega +{\omega }^2)+1[\because {\omega }^3=1]$

$=2-(-1)2-1[\because 1+\omega +{\omega }^2=0]$

$3-1=3$.