Question

If $1,\omega$ and ${\omega }^2$ are cube roots of unity, show that $(1-\omega )(1-{\omega }^2)(1-{\omega }^4)(1-{\omega }^8)=9$.

Answer 1

Given,

$(1-w)(1-w^2)(1-w^4)(1-w^8)$

$=\left[\right(1-w\left)\right(1-w^2\left)\right]\left[\right(1-w\left)\right(1-w^2\left)\right]$

$=\left[\right(1-w\left)\right(1-w^2){]}^2$

$=[w^3-w^2-w+1{]}^2$

$=[2-(w+w^2){]}^2$

$=[2-(-1){]}^2$

$=3^2$

$=9$

Hence proved.