Answer
The simplified form of a complex rational expression $\frac{\frac{a}{a-b}}{\frac{{{a}^{2}}}{{{a}^{2}}-{{b}^{2}}}}$ is $\frac{a+b}{a}$.
Work Step by Step
$\frac{\frac{a}{a-b}}{\frac{{{a}^{2}}}{{{a}^{2}}-{{b}^{2}}}}$
The numerator and denominator of the complex rational expression are single.
Divide the numerator by the denominator,
$\frac{\frac{a}{a-b}}{\frac{{{a}^{2}}}{{{a}^{2}}-{{b}^{2}}}}=\frac{a}{a-b}\div \frac{{{a}^{2}}}{{{a}^{2}}-{{b}^{2}}}$
$\begin{align}
& \frac{A}{B}\div \frac{C}{D}=\frac{A}{B}\cdot \frac{D}{C} \\
& =\frac{AD}{BC}
\end{align}$
where $\frac{A}{B},\left( B\ne 0 \right)$, $\frac{C}{D},\left( D\ne 0 \right)$ are rational expressions with $\frac{C}{D}\ne 0$
The reciprocal of $\frac{{{a}^{2}}}{{{a}^{2}}-{{b}^{2}}}$ is $\frac{{{a}^{2}}-{{b}^{2}}}{{{a}^{2}}}$
So, multiply the reciprocal of the divisor,
$\frac{\frac{a}{a-b}}{\frac{{{a}^{2}}}{{{a}^{2}}-{{b}^{2}}}}=\frac{a}{a-b}\cdot \frac{{{a}^{2}}-{{b}^{2}}}{{{a}^{2}}}$
Simplify the terms,
$\begin{align}
& \frac{\frac{a}{a-b}}{\frac{{{a}^{2}}}{{{a}^{2}}-{{b}^{2}}}}=\frac{a\cdot \left( {{a}^{2}}-{{b}^{2}} \right)}{\left( a-b \right)\cdot \left( {{a}^{2}} \right)} \\
& =\frac{a}{{{a}^{2}}}\cdot \frac{\left( {{a}^{2}}-{{b}^{2}} \right)}{\left( a-b \right)}
\end{align}$
Take the factor for the numerator term,
$\begin{align}
& \frac{\frac{a}{a-b}}{\frac{{{a}^{2}}}{{{a}^{2}}-{{b}^{2}}}}=\frac{a}{{{a}^{2}}}\cdot \frac{\left( a+b \right)\left( a-b \right)}{\left( a-b \right)} \\
& =\frac{1}{a}\cdot \left( a+b \right) \\
& =\frac{a+b}{a}
\end{align}$
