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If $\omega$ is the imaginary cube root of unity, then what is $\left(2-\omega+2 \omega^{2}\right)^{27}$ equal to?
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The correct answer is:
$-3^{27}$
Consider, $\left(2-\omega+2 \omega^{2}\right)^{27}=\left[2\left(1+\omega^{2}\right)-\omega\right]^{27}$
$=(-2 \omega-\omega)^{27}\left[\because 1+\omega+\omega^{2}=0\right]$
$=(-3 \omega)^{27}=-3^{27} \cdot \omega^{27}$
$=(-3)^{27} \cdot\left(\omega^{3}\right)^{9}=(-3)^{27}$
$=(-2 \omega-\omega)^{27}\left[\because 1+\omega+\omega^{2}=0\right]$
$=(-3 \omega)^{27}=-3^{27} \cdot \omega^{27}$
$=(-3)^{27} \cdot\left(\omega^{3}\right)^{9}=(-3)^{27}$
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