How can I solve $(1+2i)^i$.

Question

The question is how can I solve $$(1+2i)^i$$ Thanks for hints.

Answer 1

$(1+2i) = \sqrt 5 (\cos\theta + i \sin \theta)$ where $\theta = \arctan 2$

now for the tricky part!

$\sqrt 5 (\cos\theta + i \sin \theta) = e^{\sqrt 5 + i\theta}$

This is Euler's identity.

$(\sqrt 5 (\cos\theta + i \sin \theta))^i = e^{(\sqrt 5 + i\theta)^i} =e^{(-\theta + i \sqrt 5)} = e^{-\arctan 2}(\cos \sqrt 5 + i \sin \sqrt 5) $