[tex]81(\cos160^\circ+i\sin160^\circ)=81e^{160^\circ i}[/tex]
By DeMoivre's theorem,
[tex]\left(81e^{160^\circ i}\right)^{1/4}=81^{1/4}e^{(160+360k)^\circ i/4}[/tex]
where [tex]k=0,1,2,3[/tex]. [tex]81=3^4\implies81^{1/4}=3[/tex], so the 4th roots are
[tex]k=0:\quad3e^{40^\circ i}=3(\cos40^\circ+i\sin40^\circ)[/tex]
[tex]k=1:\quad3e^{130^\circ i}=3(\cos130^\circ+i\sin130^\circ)[/tex]
[tex]k=2:\quad3e^{220^\circ i}=3(\cos220^\circ+i\sin220^\circ)[/tex]
[tex]k=3:\quad3e^{310^\circ i}=3(\cos310^\circ+i\sin310^\circ)[/tex]
