Question

If $\omega$ is the cube root of unity then the value of $(1-\omega )(1-{\omega }^2)(1-{\omega }^4)(1-{\omega }^8)$ is:

• A. $9$
• B. $-9$
• C. $16$
• D. $32$

Answer 1

The correct option is A $9$

Let $\omega$ be the cube root of unity.

$\therefore {\omega }^3=1$

$(1-\omega )(1-{\omega }^2)(1-{\omega }^4)(1-{\omega }^8)=(1-\omega )(1-{\omega }^2)(1-{\omega }^3\cdot \omega )(1-{\omega }^3\cdot {\omega }^3\cdot {\omega }^2)$

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

$=(1-\omega {)}^2(1-{\omega }^2{)}^2$

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

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

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

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

$=[1+1+1{]}^2$

$=3^2$

$=9$

$\therefore (1-\omega )(1-{\omega }^2)(1-{\omega }^4)(1-{\omega }^8)=9$