The value of $(2 - \omega)\left( 2 - \omega^{2} \right)\left( 2 - \omega^{10} \right)(2 - \omega^{11})$, where $\omega$ is the complex cube root of unity, is

Question

Dr. Abubacker · Accepted Answer

(a)
$(2 - \omega)\left( 2 - \omega^{2} \right)\left( 2 - \omega^{10} \right)(2 - \omega^{11})$
$= (2 - \omega)\left( 2 - \omega^{2} \right)(2 - \omega)(2 - \omega^{2})$
$= \left\lbrack (2 - \omega)\left( 2 - \omega^{2} \right) \right\rbrack^{2}$
$= \left\lbrack 4 - 2\left( \omega{+ \omega}^{2} \right) + 1 \right\rbrack^{2} = (4 + 2 + 1)^{2} = 49$

Q. The value of $(2 - \omega)\left( 2 - \omega^{2} \right)\left( 2 - \omega^{10} \right)(2 - \omega^{11})$, where $\omega$ is the complex cube root of unity, is

Explanation

Reviewed by Dr. Abubacker

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