Principal root of 3+4i

Question

Is there a neat way of writing the principal root of 3+4i? I have an answer, but it is very ugly. Thanks for any help in advance.

Answer 1

$(2+i)^2=4-1+4i=3+4i$

You can show it by $(a+ib)^2=3+4i\implies a^2-b^2=3,2ab=4$

Polar form : $\sqrt 5 e^{i\arctan(1/2)}$

Answer 2

$3+ 4i= 5e^{ \arctan(4/3)}$. The principal $p$th root is $\sqrt[p]{5}e^{\frac{\arctan(4/3)}{p}i}$