Answer
The statement $\frac{7+3i}{5+3i}=\frac{7}{5}$ is false. The true statement is \[\frac{7+3i}{5+3i}=\frac{22}{17}-\frac{3}{17}i\].
Work Step by Step
Consider the expression, $\frac{7+3i}{5+3i}$
Multiply by the complex conjugate of the denominator in the numerator and the denominator.
$\frac{7+3i}{5+3i}=\frac{\left( 7+3i \right)}{\left( 5+3i \right)}\cdot \frac{\left( 5-3i \right)}{\left( 5-3i \right)}$
Use the FOIL method.
$\begin{align}
& \frac{7+3i}{5+3i}=\frac{\left( 7+3i \right)\left( 5-3i \right)}{\left( 5+3i \right)\left( 5-3i \right)} \\
& =\frac{35-21i+15i-9{{i}^{2}}}{25-15i+15i-9{{i}^{2}}} \\
& =\frac{35-6i-9{{i}^{2}}}{25-9{{i}^{2}}}
\end{align}$
Replace the value ${{i}^{2}}=-1$.
$\begin{align}
& \frac{7+3i}{5+3i}=\frac{35-6i-9\left( -1 \right)}{25-9\left( -1 \right)} \\
& =\frac{35-6i+9}{25+9} \\
& =\frac{44-6i}{34} \\
& =\frac{44}{34}-\frac{6}{34}i
\end{align}$
Further simply the expression.
$\frac{7+3i}{5+3i}=\frac{22}{17}-\frac{3}{17}i$
Therefore, the statement $\frac{7+3i}{5+3i}=\frac{7}{5}$ is false. The true statement is $\frac{7+3i}{5+3i}=\frac{22}{17}-\frac{3}{17}i$.
