Answer
$-16$
Work Step by Step
Step 1: $[8(\cos210^{\circ}+i\sin210^{\circ})][2(\cos330^{\circ}+i\sin330^{\circ}]$
Step 2: Using the product theorem, the expression becomes
$8(2)[\cos(210^{\circ}+330^{\circ})+i\sin(210^{\circ}+330^{\circ})]$
Step 3: Simplifying, $16(\cos540^{\circ}+i\sin540^{\circ})$
Step 4: Since $180^{\circ}$ and $540^{\circ}$ are coterminal angles, the expression becomes $16(\cos180^{\circ}+i\sin180^{\circ})$
Step 5: We know that $\cos180^{\circ}=-1$ and $\sin180^{\circ}=0$
Step 6: Substituting these values in the expression, $16(-1+(0)i)$
Step 7: Therefore, the final answer is $-16$.
