## Elementary and Intermediate Algebra: Concepts & Applications (6th Edition)

The simplified form of a complex rational expression $\frac{\frac{a}{a-b}}{\frac{{{a}^{2}}}{{{a}^{2}}-{{b}^{2}}}}$ is $\frac{a+b}{a}$.
$\frac{\frac{a}{a-b}}{\frac{{{a}^{2}}}{{{a}^{2}}-{{b}^{2}}}}$ 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}