Answer
$$ - 3.7588 - 1.3681i$$
Work Step by Step
$$\eqalign{
& {\left[ {2\operatorname{cis} \frac{{5\pi }}{9}} \right]^2} \cr
& {\text{Write in trigonometric form}} \cr
& = {\left[ {2\left( {\cos \frac{{5\pi }}{9} + i\sin \frac{{2\pi }}{5}} \right)} \right]^2} \cr
& {\text{Use the power property}} \cr
& {\text{ = }}{{\text{2}}^2}{\left( {\cos \frac{{5\pi }}{9} + i\sin \frac{{2\pi }}{5}} \right)^2} \cr
& = 4\left[ {\cos 2\left( {\frac{{5\pi }}{9}} \right) + i\sin 2\left( {\frac{{2\pi }}{5}} \right)} \right] \cr
& = 4\left( {\cos \frac{{10\pi }}{9} + i\sin \frac{{10\pi }}{9}} \right) \cr
& {\text{Use a calculator }}\left( {{\text{radian mode}}} \right) \cr
& = - 3.7588 - 1.3681i \cr} $$
