Question #9daa9

Oct 28, 2016

Answer:

$\textcolor{g r e e n}{\frac{1}{{a}^{2} b + 1}}$

Explanation:

If we temporarily replace ${a}^{2} b$ with $x$

$\frac{{a}^{2} b - 1}{{a}^{4} {b}^{2} - 1} \rightarrow \frac{x - 1}{{x}^{2} - 1}$

We know that ${x}^{2} - 1$ can be factored as $\left(x - 1\right) \left(x + 1\right)$

So $\frac{x - 1}{{x}^{2} - 1} = \frac{\cancel{x - 1}}{\left(\cancel{x - 1}\right) \left(x + 1\right)} = \frac{1}{x + 1}$

Restoring $x$ back to ${a}^{2} b$, we have
$\textcolor{w h i t e}{\text{XXX}} \frac{{a}^{2} b - 1}{{a}^{4} {b}^{2} - 1} = \frac{1}{{a}^{2} b + 1}$