Question

If $1,\omega and{\omega }^2$ are the cube roots of $1$ then $\left(1+\omega \right)\left(1+{\omega }^2\right)\left(1+{\omega }^4\right)\left(1+{\omega }^8\right)$ equals

• A. $1$
• B. $0$
• C. $-1$
• D. $2$

Answer 1

The correct option is A

$1$

The explanation of the correct answer:

Step1. Define the cube roots of unity:

Given,

$1,\omega ,and{\omega }^2$are cube roots of unity.

$$\begin{gathered} 1+\omega +{\omega }^2=0\_\_\_\_\_\_\_\_\left(1\right)\left[\mathbf{from}\mathbf{}\mathbf{definition}\mathbf{}\mathbf{,}\mathbf{}\mathbf{\omega }\mathbf{=}\frac{\mathbf{-}\mathbf{1}\mathbf{+}\mathbf{i}\sqrt{\mathbf{3}}}{\mathbf{2}}\mathbf{,}\mathbf{}{\mathbf{\omega }}^{\mathbf{2}}\mathbf{=}\frac{\mathbf{-}\mathbf{1}\mathbf{-}\mathbf{i}\sqrt{\mathbf{3}}}{\mathbf{2}}\right] \\ 1\times \omega \times {\omega }^2=1\_\_\_\_\_\_\_\_\left(2\right) \end{gathered}$$

Step2. Find the value of $\left(1+\omega \right)\left(1+{\omega }^2\right)\left(1+{\omega }^4\right)\left(1+{\omega }^8\right)$:

$$\begin{gathered} \left(1+\omega \right)\left(1+{\omega }^2\right)\left(1+{\omega }^4\right)\left(1+{\omega }^8\right) \\ =\left(-{\omega }^2\right)\left(-\omega \right)\left[1+\left({\omega }^3\right)\left(\omega \right)\right]\left[1+{\left({\omega }^3\right)}^2\left({\omega }^2\right)\right]\left[\mathbf{Substitute}\mathbf{}\mathbf{the}\mathbf{}\mathbf{value}\mathbf{}\mathbf{from}\mathbf{\left(}\mathbf{1}\mathbf{\right)}\mathbf{}\mathbf{and}\mathbf{}\mathbf{\left(}\mathbf{2}\mathbf{\right)}\right] \\ =\left({\omega }^3\right)\left[1+\left(1\right)\omega \right]\left[1+{\left(1\right)}^2\left({\omega }^2\right)\right] \\ =\left(1\right)\left(-{\omega }^2\right)\left(-\omega \right) \\ ={\omega }^3 \\ =\mathbf{1} \end{gathered}$$

Hence, Option(A) is the correct answer.