Answer
$-\frac{21\sqrt 3}{2}-\frac{21}{2}i$
Work Step by Step
First, we use the product theorem to multiply the absolute values and add the arguments:
$(3$ cis $300^{\circ})(7$ cis $270^{\circ})$
$=3(7)$ cis $(300^{\circ}+270^{\circ})$
$=21$ cis $(570^{\circ})$
Next, we change the expression into its equivalent form:
$=21$ cis $(570^{\circ})$
$=21 (\cos 570^{\circ}+i\sin 570^{\circ})$
Since we know that $\cos 570^{\circ}=\cos 210^{\circ}=-\frac{\sqrt 3}{2}$ and $\sin 570^{\circ}=\sin 210^{\circ}=-\frac{1}{2}$, we subsitute these values in the expression and simplify:
$21 (\cos 570^{\circ}+i\sin 570^{\circ})$
$=21 [-\frac{\sqrt 3}{2}+i(-\frac{1}{2})]$
$=-\frac{21\sqrt 3}{2}-\frac{21}{2}i$