For $z, w \in \mathbb{C}$ define the principal value of $z^{w}$. State de Moivre's theorem.

Hence solve the equations (i) $z^{6}=\sqrt{3}+i$, (ii) $z^{1 / 6}=\sqrt{3}+i$, (iii) $i^{z}=\sqrt{3}+i$ (iv) $\left(e^{5 i \pi / 2}\right)^{z}=\sqrt{3}+i$

[In each expression, the principal value is to be taken.]