[tex]z = 1+i \\ \\ z^{2019} = (1+i)^{2019} = \Big[(1+i)^2\Big]^{1009}\cdot (1+i) = \\ \\ =(2i)^{1009}\cdot (1+i) =2^{1009}\cdot i^{1009}\cdot (1+i) = \\ \\ =2^{1009}\cdot {(i^2)}^{504}\cdot i\cdot (1+i) =2^{1009}\cdot (-1)^{504}\cdot i\cdot(1+i) =\\ \\=2^{1009}\cdot 1\cdot i\cdot(1+i)=2^{1009}(i-1) =\boxed{-2^{1009}+2^{1009}i}[/tex]
