Question

If $\omega$ is a complex cube root of unity, then the value of $(1+\omega )(1+{\omega }^2)(1+{\omega }^4)(1+{\omega }^8)...(2n+1)$ terms is

• A. $1$
• B. $2\omega$
• C. $-\omega$
• D. $-{\omega }^2$

Answer 1

Answer: (C)

$-\omega$

We know,

$1+\omega +{\omega }^2=0$

$\Rightarrow (1+\omega )(1+{\omega }^2)(1+{\omega }^4)(1+{\omega }^8)...(2n+1)$ terms

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

$=({\omega }^3{)}^n\times (-\omega )$

$=-\omega$