Question

If $\omega$ is an imaginary cube root of unity, then the value of $(1+\omega )(1+{\omega }^2)(1+{\omega }^3)(1+{\omega }^4)(1+{\omega }^5)....(1+{\omega }^{3n})$ is

• A. $2^{3n}$
• B. $2^{2n}$
• C. $2^n$
• D. None of these

Answer 1

The correct option is C $2^n$

We have,
$(1+\omega )(1+{\omega }^2)(1+{\omega }^3)(1+{\omega }^4)....(1+{\omega }^{3n})$
$=\left[\right\{(1+\omega )(1+{\omega }^2)\left\}\right\{(1+{\omega }^4)(1+{\omega }^5)\}....\{(1+{\omega }^{3n-2})(1+{\omega }^{3n-1})\times (1+{\omega }^3)(1+{\omega }^9)....(1+\omega )$
$=2^n\left[\right\{(-{\omega }^2)(-\omega )\left\}\right\{(-{\omega }^2)(-\omega )]....\{(-{\omega }^2)(-\omega )\left\}\right]$
$\Rightarrow \left[\right(1+\omega \left)\right(1+{\omega }^2\left)\right\{(1+\omega )(1+{\omega }^2)\left\}\right\{(1+\omega )(1+\omega )\}....\{(1+\omega )(1+{\omega }^2)\left\}\right]\times (1+1)(1+1)....(1+1)$
$=2^n$