Answer
The standard form of the expression$\frac{5}{2-i}$ is $2+i$.
Work Step by Step
Consider the expression,
$\frac{5}{2-i}$
Since, the imaginary part is in the denominator, we multiply the numerator and denominator by the complex conjugate of the denominator -- that is, for the complex number $\left( 2-i \right)$, its complex conjugate is $\left( 2+i \right)$
Multiply the expression by $\frac{\left( 2+i \right)}{\left( 2+i \right)}$.
$\begin{align}
& \frac{5}{2-i}=\frac{5}{2-i}\cdot \frac{2+i}{2+i} \\
& =\frac{5\left( 2+i \right)}{\left( 2-i \right)\left( 2+i \right)}
\end{align}$
The product of the complex number $\left( a+bi \right)$ and its complex conjugate $\left( a-bi \right)$ results in a real number -- that is, $\left( a+bi \right)\left( a-bi \right)={{a}^{2}}+{{b}^{2}}$
$\begin{align}
& \frac{5\left( 2+i \right)}{\left( 2-i \right)\left( 2+i \right)}=\frac{5\left( 2+i \right)}{{{2}^{2}}+{{1}^{2}}} \\
& =\frac{5\left( 2+i \right)}{4+1}
\end{align}$
Further simplify the expression.
$\begin{align}
& \frac{5}{2-i}=\frac{5\left( 2+i \right)}{4+1} \\
& =\frac{5\left( 2+i \right)}{5} \\
& =2+i
\end{align}$
Hence, the standard form of the expression $\frac{5}{2-i}$ is $2+i$.
