Question

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

• A. $3$
• B. $2$
• C. $1$
• D. none of these

Answer 1

Answer: (A)

$3$

Now,

$(1-\omega )(1-{\omega }^2)(1+{\omega }^4)(1+{\omega }^8)$

$=(1-\omega -{\omega }^2+{\omega }^3)(1+\omega )(1+{\omega }^2)$ [ Since ${\omega }^3=1$ then ${\omega }^4=\omega$ and ${\omega }^8={\omega }^2$]

$=(2-\omega -{\omega }^2)(1+\omega +{\omega }^2+{\omega }^3)$

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

$=3.1$ [ As $1+\omega +{\omega }^2=0\Rightarrow -\omega -{\omega }^2=1$]