[tex]x^3-27i=0[/tex]
[tex]x^3=27i[/tex]
[tex]x^3=27e^{i\pi/2}[/tex]
[tex]x=\left(27e^{i\pi/2}\right)^{1/3}[/tex]
[tex]x=3e^{i(\pi/2+2\pi k)/3}[/tex]
[tex]x=3e^{i\pi(4k+1)/6}[/tex]
where [tex]k\in\{0,1,2\}[/tex]. This means you have
[tex]x=3e^{i\pi/6}=\dfrac32(\sqrt3+i)[/tex]
[tex]x=3e^{i5\pi/6}=\dfrac32(-\sqrt3+i)[/tex]
[tex]x=3e^{i9\pi/6}=-3i[/tex]
as the solutions to the original equation.
