Answer
$$ - 511.3793 - 295.2450i$$
Work Step by Step
$$\eqalign{
& {\left[ {24.3\operatorname{cis} \frac{{7\pi }}{{12}}} \right]^2} \cr
& {\text{Write in trigonometric form}} \cr
& = {\left[ {24.3\left( {\cos \frac{{7\pi }}{{12}} + i\sin \frac{{7\pi }}{{12}}} \right)} \right]^2} \cr
& {\text{Use the power property}} \cr
& {\text{ = 24}}{\text{.}}{{\text{3}}^2}{\left( {\cos \frac{{7\pi }}{{12}} + i\sin \frac{{7\pi }}{{12}}} \right)^2} \cr
& = 590.49\left[ {\cos 2\left( {\frac{{7\pi }}{{12}}} \right) + i\sin 2\left( {\frac{{7\pi }}{{12}}} \right)} \right] \cr
& = 590.49\left( {\cos \frac{{14\pi }}{{12}} + i\sin \frac{{14\pi }}{{12}}} \right) \cr
& {\text{Use a calculator }}\left( {{\text{radian mode}}} \right) \cr
& = - 511.3793 - 295.2450i \cr} $$
