If $\omega$ is a cube root of unity and $(1 + \omega)^7 = A + B\omega$, then $A + B$ equals:

We know $\omega^3 = 1$ and $1 + \omega + \omega^2 = 0$. So $1 + \omega = -\omega^2$. Hence $(1 + \omega)^7 = (-\omega^2)^7 = (-1)^7 \omega^{14} = -\omega^{14}$. Now $\omega^{14} = \omega^{3 \times 4 + 2} = \omega^2$. Therefore $(1 + \omega)^7 = -\omega^2 = 1 + \omega$. So comparing with $A + B\omega$, we have $A = 1$, $B = 1$. $\Rightarrow A + B = 2.$