Find the square roots of the complex number $z = 3 + 4i$.

Express the complex number $z = 1 + i\sqrt{3}$ in polar form $r(\cos\theta + i\sin\theta)$ and find its principal argument.

Find the modulus and principal argument of $z = -1 + i\sqrt{3}$, and express it in polar and exponential forms.

If $\omega$ and $\omega^2$ are the complex cube roots of unity, prove that: (a) $1 + \omega + \omega^2 = 0$ (b) $\omega^3 = 1$ (c) $(1 - \omega + \omega^2)(1 + \omega - \omega^2) = 4$