College Algebra, Chapter 1, 1.4, Section 1.4, Problem 74

Suppose that $z = a + bi$ and $w = c + di$, then the symbol $\overline{z}$ represents the complex conjugate of $z$. Prove $\overline{zw} = \overline{z} \cdot \overline{w}$


$
\begin{equation}
\begin{aligned}

\overline{zw} =& \overline{(a + bi)(c + di)}
&& \text{Model}
\\
\\
=& \overline{(ac + adi + cbi + bdi^2)}
&& \text{Use FOIL method}
\\
\\
=& \overline{(ac + adi + cbi + bd(-1))}
&& \text{Recall that } i^2 = -1
\\
\\
=& \overline{(ac - bd) + (ad + cb)i}
&& \text{Apply complex conjugate}
\\
\\
=& (ac - bd) - (ad + cb)i
&&
\\
\\
\overline{z} \cdot \overline{w} =& \overline{a + bi} \cdot \overline{c + di}
&& \text{Apply complex conjugate}
\\
\\
=& (a - bi) \cdot (c - di)
&& \text{Apply FOIL method}
\\
\\
=& ac - adi - cbi + (bdi^2)
&& \text{Recall that } i^2 = -1
\\
\\
=& ac - adi - cbi + (bd(-1))
&& \text{Simplify}
\\
\\
=& (ac - bd) - (ad + cb) i
&&



\end{aligned}
\end{equation}
$