Answer
The standard form of the expression $\frac{2+3i}{2+i}$ is $\frac{7}{5}+\frac{4}{5}i$.
Work Step by Step
Consider the expression,$\frac{2+3i}{2+i}$
Multiply by the complex conjugate of the denominator in the numerator and the denominator.
$\frac{2+3i}{2+i}=\frac{\left( 2+3i \right)}{\left( 2+i \right)}\cdot \frac{\left( 2-i \right)}{\left( 2-i \right)}$
Use the FOIL method.
\[\begin{align}
& \frac{2+3i}{2+i}=\frac{\left( 2+3i \right)\left( 2-i \right)}{\left( 2+i \right)\left( 2-i \right)} \\
& =\frac{4-2i+6i-3{{i}^{2}}}{4-2i+2i-{{i}^{2}}} \\
& =\frac{4+4i-3{{i}^{2}}}{4-{{i}^{2}}}
\end{align}\]
Replace the value ${{i}^{2}}=-1$.
\[\begin{align}
& \frac{2+3i}{2+i}=\frac{4+4i-3\left( -1 \right)}{4-\left( -1 \right)} \\
& =\frac{4+4i+3}{4+1}
\end{align}\]
Make a group of real and imaginary terms.
\[\begin{align}
& \frac{2+3i}{2+i}=\frac{\left( 4+3 \right)+4i}{5} \\
& =\frac{7+4i}{5}
\end{align}\]
Express the complex number in the standard form.
\[\frac{2+3i}{2+i}=\frac{7}{5}+\frac{4}{5}i\]
Therefore, the standard form of the expression $\frac{2+3i}{2+i}$ is $\frac{7}{5}+\frac{4}{5}i$.
