A New Meaning to Complex Roots

$\begin{aligned} \sqrt[1-i] k & = k^\frac 1{1-i} = e^{\frac {\ln k}{1-i}} = e^{\frac {\ln k}2(1+i)} \\ & = e^{\frac {\ln k}2} \cdot \blue{e^{\frac {\ln 2}2i}} & \small \blue{\text{By Euler's formula}} \\ & = e^{\frac {\ln k}2} \blue{\left(\cos \frac {\ln 2}2 + i \sin \frac {\ln 2}2\right)} \end{aligned}$

Therefore, $\sqrt[1-i] k$ is real when $\ln k = 2n\pi$ , where $n$ is an integer. For $k \ne 1$ , the smallest positive value of $\sqrt[1-i] k$ occurs when $\ln k = 4\pi$ , and $\min \left \lfloor \sqrt[1-i] k \right \rfloor = \lfloor e^{2\pi} \rfloor = \boxed{535}$ .