$z^{6} = \frac{1+i}{1-i} \Rightarrow z^6= \frac{(1+i)^2}{(1-i)(1+i)}= \frac{1^1+2i+i^2}{1^2-i^2}= \frac{1+2i-1}{1-(-1)} = \frac{2i}{2}=i=Cos \frac{ \pi }{2} +iSin \frac{ \pi }{2}$
$z_k= \sqrt[6]{ | i | } (Cos \frac{ \frac{ \pi }{2} +2k \pi }{6} +iSin \frac{ \frac{ \pi }{2} +2k \pi }{6})= \sqrt[6]{1} (Cos \frac{(4k+1) \pi }{12} +iSin \frac{(4k+1) \pi }{12} )$
$=Cos \frac{(4k+1) \pi }{12} +iSin \frac{(4k+1) \pi }{12} \wedge k=0,1...5$
$ \Box $
