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.

Express the following in the standard form $x + iy$ ($x, y \in \mathbb{R}$): (a) $\frac{3 + 2i}{2 - 5i}$ (b) $(1 + i)^4$ (c) $\frac{i^{4n+3} + i^{4n+1}}{i^{4n+2}}$ for any $n \in \mathbb{Z}$

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) $(1 - \omega + \omega^2)(1 + \omega - \omega^2) = 4$