x^2=sqrt(3)/2 + i/2=cos 60+jsin 60 (one of 2 options)

x^4= cos 120+jsin 120 (numbers of degrees)

x=cos 30 +jsin 30 (one of 2 options)

x^5= x^4*x= cos 150 + jsin 150 =A

1/x= cos 30 - jsin 30=B

A+B= -COS 30 +COS 30 +JSIN 30-JSIN 30=0

QED

I did not bother to show all 4 possibilities...