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If $\omega$ is a complex cube root of unity, then what is $\omega^{10}+\omega^{-10}$ equal to ?
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The correct answer is:
-1
Consider $\omega^{10}+\omega^{-10}=\omega^{10}+\frac{1}{\omega^{10}}$
$=\left(\omega^{3}\right)^{3} \cdot \omega+\frac{1}{\left(\omega^{3}\right)^{3} \cdot \omega}=\omega+\frac{1}{\omega}=-1$
$=\left(\omega^{3}\right)^{3} \cdot \omega+\frac{1}{\left(\omega^{3}\right)^{3} \cdot \omega}=\omega+\frac{1}{\omega}=-1$
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