Answer with explanation:
The given expression which is in complex form is :
=81 (Cos 320°+Sin 320°)------------------------------------(1)
For, a Complex number in the form of
Z=r [Cos A + i Sin A], Can be written as

We have to find four roots of expression (1).
![Z^4=81 (Cos 320^{\circ}+iSin 320^{\circ})\\\\Z=[81\times (Cos(2k\pi + 320^{\circ})+iSin (2k\pi +320^{\circ})]^{\frac{1}{4}}\\\\Z={3^{{4}\times^{\frac{1}{4}}}\times e^{i(\frac{2k\pi +320^{\circ}}{4})}}} \\\\Z=3e^{i(\frac{k\pi}{2}+ 80^{\circ}})\\\\Z_{0}=3(Cos 80^{\circ}+iSin 80^{\circ})\\\\Z_{1}=3(Cos 170^{\circ}+iSin 170^{\circ})\\\\Z_{2}=3(Cos 260^{\circ}+iSin260^{\circ})\\\\Z_{3}=3(Cos 350^{\circ}+iSin 350^{\circ})](https://tex.z-dn.net/?f=Z%5E4%3D81%20%28Cos%20320%5E%7B%5Ccirc%7D%2BiSin%20320%5E%7B%5Ccirc%7D%29%5C%5C%5C%5CZ%3D%5B81%5Ctimes%20%28Cos%282k%5Cpi%20%2B%20320%5E%7B%5Ccirc%7D%29%2BiSin%20%282k%5Cpi%20%2B320%5E%7B%5Ccirc%7D%29%5D%5E%7B%5Cfrac%7B1%7D%7B4%7D%7D%5C%5C%5C%5CZ%3D%7B3%5E%7B%7B4%7D%5Ctimes%5E%7B%5Cfrac%7B1%7D%7B4%7D%7D%7D%5Ctimes%20e%5E%7Bi%28%5Cfrac%7B2k%5Cpi%20%2B320%5E%7B%5Ccirc%7D%7D%7B4%7D%29%7D%7D%7D%20%5C%5C%5C%5CZ%3D3e%5E%7Bi%28%5Cfrac%7Bk%5Cpi%7D%7B2%7D%2B%2080%5E%7B%5Ccirc%7D%7D%29%5C%5C%5C%5CZ_%7B0%7D%3D3%28Cos%2080%5E%7B%5Ccirc%7D%2BiSin%2080%5E%7B%5Ccirc%7D%29%5C%5C%5C%5CZ_%7B1%7D%3D3%28Cos%20170%5E%7B%5Ccirc%7D%2BiSin%20170%5E%7B%5Ccirc%7D%29%5C%5C%5C%5CZ_%7B2%7D%3D3%28Cos%20260%5E%7B%5Ccirc%7D%2BiSin260%5E%7B%5Ccirc%7D%29%5C%5C%5C%5CZ_%7B3%7D%3D3%28Cos%20350%5E%7B%5Ccirc%7D%2BiSin%20350%5E%7B%5Ccirc%7D%29)
The four values are obtained for, k=0,1,2,3,.