Find the 4th roots of the equation below. Convert the answers to rectangular form in radians. 24 = 81 (cos+ i sin 공)

 $$\begin{gathered} z^4=81(\cos \frac{\mathrm{\pi }}{6}+i\sin \frac{\mathrm{\pi }}{6}) \\ \Rightarrow z={\left[81\left(\cos \right(2k\mathrm{\pi }+\frac{\mathrm{\pi }}{6})+i\sin (2k\mathrm{\pi }+\frac{\mathrm{\pi }}{6})\right]}^{\frac{1}{4}} \\ =\left[{\left(81\right)}^{\frac{1}{4}}\left(\cos \frac{1}{4}\right(2k\mathrm{\pi }+\frac{\mathrm{\pi }}{6})+i\sin \frac{1}{4}(2k\mathrm{\pi }+\frac{\mathrm{\pi }}{6})\right],k=0,1,2,3 \\ =\left[3\left(\cos \frac{1}{4}\right(2k\mathrm{\pi }+\frac{\mathrm{\pi }}{6})+i\sin \frac{1}{4}(2k\mathrm{\pi }+\frac{\mathrm{\pi }}{6})\right],k=0,1,2,3 \\ =3\left[\cos \left(\frac{\mathrm{\pi }}{24}\right)+\mathrm{isin}\left(\frac{\mathrm{\pi }}{24}\right)\right],for k=0 \\ =3\left[\cos \left(\frac{13\mathrm{\pi }}{24}\right)+\mathrm{isin}\left(\frac{13\mathrm{\pi }}{24}\right)\right],for k=1 \\ =3\left[\cos \left(\frac{25\mathrm{\pi }}{24}\right)+\mathrm{isin}\left(\frac{25\mathrm{\pi }}{24}\right)\right],for k=2 \\ =3\left[\cos \left(\frac{37\mathrm{\pi }}{24}\right)+\mathrm{isin}\left(\frac{37\mathrm{\pi }}{24}\right)\right],for k=3 \end{gathered}$$