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 $\ln(a+bi)$.
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Logarithms of complex numbers are more complicated than for real numbers. They are intrinsically multi-valued, and to get a well defined function, you need to choose a "branch cut." – Cheerful Parsnip Apr 05 '18 at 17:29
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@LatinWolf Please remember that you can choose an answer among the given if the OP is solved, more details here https://meta.stackexchange.com/questions/5234/how-does-accepting-an-answer-work – user Apr 07 '18 at 20:06
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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 $k=0,...,n-1$.
Adrian Keister
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I guess $k=0,n-1$ is computer-speak for what in mathematics we write $k=0,\dots,n-1$. – GEdgar Apr 05 '18 at 17:46
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