Clear [z]

For[k=1,k<4,k++,x[k-1]=(10^(1/6))*Cos[(ArcTan[1/3]+k π)/3];y[k-1]=(10^(1/6))*Sin[(ArcTan[1/3]+k π)/3]; z[k]=x[k-1]+I*y[k-1]]

z[1]//Simplify
z[2]//Simplify
z[3]//Simplify