College Algebra, Chapter 1, 1.4, Section 1.4, Problem 28

Evaluate the expression $\displaystyle \left( \frac{2}{3} + 12i \right) \left( \frac{1}{6} + 24i \right)$ in the form of $a + bi$.


$
\begin{equation}
\begin{aligned}
&= \left( \frac{2}{3} + 12i \right) \left( \frac{1}{6} + 24i \right)\\
\\
&= \left( \frac{2}{3} \right) \left( \frac{1}{6} \right) + \frac{2}{3} (64i) + 12i \left( \frac{1}{6} \right) + (12i) (24i) && \text{Use FOIL method}\\
\\
&= \frac{1}{9} + \frac{128i}{3} + 2i + 288 i^2 && \text{Evaluate}\\
\\
&= \frac{1}{9} + \frac{128i}{3} + 2i + 288 (-1) && \text{recall that } i^2 = -1\\
\\
&= \left( \frac{1}{9} - 288 \right) + \left( \frac{128}{3} + 2 \right)i && \text{Combine like terms}\\
\\
&= \frac{-2591}{9} + \frac{134}{3}i && \text{Simplify}
\end{aligned}
\end{equation}
$