We know that the product of two complex numbers follows the property:
[tex]r_1cis(\theta_1)\cdot r_2cis(\theta_2)=r_1\cdot r_2\cdot cis(\theta_1+\theta_2)[/tex]
From the problem, we identify:
[tex]\begin{gathered} r_1=3\sqrt{3} \\ \theta_1=\frac{\pi}{8} \\ r_2=3\sqrt{5} \\ \theta_2=\frac{2\pi}{3} \end{gathered}[/tex]
Then, using the product property:
[tex]\begin{gathered} 3\sqrt{3}cis(\frac{\pi}{8})\cdot3\sqrt{5}cis(\frac{2\pi}{3})=3\sqrt{3}\cdot3\sqrt{5}\cdot cis(\frac{\pi}{8}+\frac{2\pi}{3}) \\ \\ \Rightarrow3\sqrt{3}cis(\frac{\pi}{8})\cdot3\sqrt{5}cis(\frac{2\pi}{3})=9\sqrt{15}cis(\frac{19\pi}{24}) \end{gathered}[/tex]
To the nearest hundredth, this is:
[tex]Answer:34.86cis(2.49)[/tex]
