Q.If $1, \omega, \omega^2$ are the cube roots of unity, then find the value of $(1-\omega+\omega^2)^5 + (1+\omega-\omega^2)^5$.

Use $1+\omega+\omega^2=0$ to rewrite each bracket as $-2\omega$ and $-2\omega^2$, then reduce the powers using $\omega^3=1$.

Since $1,\omega,\omega^2$ are the cube roots of unity, $1+\omega+\omega^2=0$ and $\omega^3=1$.

From $1+\omega+\omega^2=0$: $1+\omega^2=-\omega$, so

$1-\omega+\omega^2 = (1+\omega^2)-\omega = -\omega-\omega = -2\omega.$

Also $1+\omega=-\omega^2$, so

$1+\omega-\omega^2 = (1+\omega)-\omega^2 = -\omega^2-\omega^2 = -2\omega^2.$

Hence

$(1-\omega+\omega^2)^5+(1+\omega-\omega^2)^5 = (-2\omega)^5+(-2\omega^2)^5 = -32\omega^5-32\omega^{10}.$

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