Finding a general formula for $\sqrt[n]{a+bi}$.

Question

So $\sqrt[n]{a+bi}$ can be written as $$\exp\left(\dfrac{\ln(a+bi)}{n}\right).$$ However I don't know how to continue since I don't know a general rule for $\ln(a+bi)$.

Answer 1

Since $a+bi$ can be written in exponential form $re^{i(\theta+2k\pi)}$ we have

$$\sqrt[n]{a+bi}=\sqrt[n]r\,\exp\left[i\left(\frac{\theta}n+\frac{2k\pi}n\right)\right]$$

for $k=0,...,n-1$.