Question

If $1,\omega and{\omega }^2$are the cube roots of unity, then $\left(1-\omega +{\omega }^2\right)\left(1-{\omega }^2+{\omega }^4\right)\left(1-{\omega }^4+{\omega }^8\right)\left(1-{\omega }^8+{\omega }^{16}\right)......upto2n$ factors is

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

Answer 1

The correct option is B

$2^{2n}$

Explanation of the correct option:

Step1. Define the cube root of unity:

Given,

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

factor$$\begin{gathered} 1+\omega +{\omega }^2=0\_\_\_\_\left(1\right)\left[\mathit{F}\mathit{r}\mathit{o}\mathit{m}\mathbf{}\mathit{d}\mathit{e}\mathit{f}\mathit{i}\mathit{n}\mathit{i}\mathit{t}\mathit{i}\mathit{o}\mathit{n}\mathbf{}\mathit{\omega }\mathbf{=}\frac{\mathbf{-}\mathbf{1}\mathbf{+}\mathbf{i}\sqrt{\mathbf{3}}}{\mathbf{2}}\mathbf{,}\mathbf{}{\mathit{\omega }}^{\mathbf{2}}\mathbf{=}\frac{\mathbf{-}\mathbf{1}\mathbf{-}\mathbf{i}\sqrt{\mathbf{3}}}{\mathbf{2}}\right] \\ 1\times \omega \times {\omega }^2=1\_\_\_\_\_\left(2\right) \end{gathered}$$

Step2. Find the value of each term $\mathbf{(}\mathbf{1}\mathbf{-}\mathbf{\omega }\mathbf{+}{\mathbf{\omega }}^{\mathbf{2}}\mathbf{)}\mathbf{(}\mathbf{1}\mathbf{-}{\mathbf{\omega }}^{\mathbf{2}}\mathbf{+}{\mathbf{\omega }}^{\mathbf{4}}\mathbf{)}\mathbf{(}\mathbf{1}\mathbf{-}{\mathbf{\omega }}^{\mathbf{4}}\mathbf{+}{\mathbf{\omega }}^{\mathbf{8}}\mathbf{)}\mathbf{(}\mathbf{1}\mathbf{-}{\mathbf{\omega }}^{\mathbf{8}}\mathbf{+}{\mathbf{\omega }}^{\mathbf{16}}\mathbf{)}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{}\mathit{u}\mathit{p}\mathbf{}\mathit{t}\mathit{o}\mathbf{}\mathbf{2}\mathit{n}$ factor:

$\begin{array}{rcl}\left(1-\omega +{\omega }^2\right)& =& 1+{\omega }^2-\omega \\ & =& -\omega -\omega \left[\mathit{F}\mathit{r}\mathit{o}\mathit{m}\mathbf{\left(}\mathbf{1}\mathbf{\right)}\mathbf{,}\mathbf{}\mathbf{}\mathbf{1}\mathbf{+}{\mathit{\omega }}^{\mathbf{2}}\mathbf{=}\mathbf{-}\mathit{\omega }\right]\\ & =& -2\omega \\ \left(1-{\omega }^2+{\omega }^4\right)& =& 1-{\omega }^2+\left({\omega }^3\times \omega \right)\\ & =& 1-{\omega }^2+\left(1\times \omega \right)\mathbf{\left[}\mathbf{F}\mathbf{r}\mathbf{o}\mathbf{m}\mathbf{\left(}\mathbf{2}\mathbf{\right)}\mathbf{}\mathbf{,}\mathbf{}\left({\omega }^3=1\right)\mathbf{\right]}\\ & =& 1-{\omega }^2+\omega \\ & =& -{\omega }^2-{\omega }^2\mathbf{\left[}\mathbf{F}\mathbf{r}\mathbf{o}\mathbf{m}\mathbf{\left(}\mathbf{1}\mathbf{\right)}\mathbf{\right]}\\ & =& -2{\omega }^2\\ \left(1-{\omega }^4+{\omega }^8\right)& =& 1-{\left(\omega \right)}^3\left(\omega \right)+{\left({\omega }^3\right)}^2\left({\omega }^2\right)\\ & =& 1-\omega +{\omega }^2\mathbf{\left[}\mathbf{F}\mathbf{r}\mathbf{o}\mathbf{m}\mathbf{\left(}\mathbf{2}\mathbf{\right)}\mathbf{,}\mathbf{}{\mathbf{\omega }}^{\mathbf{3}}\mathbf{=}\mathbf{1}\mathbf{\right]}\\ & =& -\omega -\omega \mathbf{\left[}\mathbf{F}\mathbf{r}\mathbf{o}\mathbf{m}\mathbf{\left(}\mathbf{1}\mathbf{\right)}\mathbf{}\mathbf{,}\mathbf{}\mathbf{1}\mathbf{+}{\mathbf{\omega }}^{\mathbf{2}}\mathbf{=}\mathbf{-}\mathbf{\omega }\mathbf{\right]}\\ & =& -2\omega \\ \left(1-{\omega }^8+{\omega }^{16}\right)& =& 1-{\left({\omega }^3\right)}^2\left({\omega }^2\right)+{\left({\omega }^3\right)}^5\left(\omega \right)\\ & =& 1-{\omega }^2+\omega \\ & =& -{\omega }^2-{\omega }^2\\ & =& -2{\omega }^2.......soonupto2nfactors\end{array}$

Step3. Find the value of $\mathbf{(}\mathbf{1}\mathbf{-}\mathbf{\omega }\mathbf{+}{\mathbf{\omega }}^{\mathbf{2}}\mathbf{)}\mathbf{(}\mathbf{1}\mathbf{-}{\mathbf{\omega }}^{\mathbf{2}}\mathbf{+}{\mathbf{\omega }}^{\mathbf{4}}\mathbf{)}\mathbf{(}\mathbf{1}\mathbf{-}{\mathbf{\omega }}^{\mathbf{4}}\mathbf{+}{\mathbf{\omega }}^{\mathbf{8}}\mathbf{)}\mathbf{(}\mathbf{1}\mathbf{-}{\mathbf{\omega }}^{\mathbf{8}}\mathbf{+}{\mathbf{\omega }}^{\mathbf{16}}\mathbf{)}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{}\mathbf{up}\mathbf{}\mathbf{to}\mathbf{}\mathbf{2}\mathbf{n}\mathbf{}\mathbf{factors}\mathbf{:}$

$$\begin{gathered} (1-\mathrm{\omega }+{\mathrm{\omega }}^2)(1-{\mathrm{\omega }}^2+{\mathrm{\omega }}^4)(1-{\mathrm{\omega }}^4+{\mathrm{\omega }}^8)(1-{\mathrm{\omega }}^8+{\mathrm{\omega }}^{16})......upto2nfactors \\ ={\left(-2\omega \right)}^n{\left(-2{\omega }^2\right)}^n\mathbf{[}\mathbf{Substitute}\mathbf{}\mathbf{the}\mathbf{}\mathbf{value}\mathbf{}\mathbf{of}\mathbf{}\mathbf{required}\mathbf{}\mathbf{terms}\mathbf{]} \\ ={\left(4\omega {\omega }^2\right)}^n \\ ={\left(4{\omega }^3\right)}^n\left[\mathbf{\because }{\mathbf{\omega }}^{\mathbf{3}}\mathbf{=}\mathbf{1}\right] \\ ={\left(4\right)}^n \\ =2^{2\mathrm{n}}\mathbf{} \end{gathered}$$

Hence, Option (B) is the correct answer.