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

Using $1+\omega+\omega^2=0$, both brackets simplify to $-2\omega$ and $-2\omega^2$; raising to the 5th power and using $\omega^3=1$ gives the value $32$.

Step 1 — Simplify each bracket using $1+\omega+\omega^2=0$.

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

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

Step 2 — Raise to the 5th power.

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

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

Step 3 — Reduce powers of $\omega$ using $\omega^3=1$. …