Answer
$\frac{x}{a}$
Work Step by Step
$\frac{x-{{a}^{-1}}}{a-{{x}^{-1}}}$
$\frac{x-{{a}^{-1}}}{a-{{x}^{-1}}}=\frac{x-\frac{1}{a}}{a-\frac{1}{x}}$
$\begin{align}
& \frac{x-\frac{1}{a}}{a-\frac{1}{x}}=\frac{\left( x-\frac{1}{a} \right)ax}{\left( a-\frac{1}{x} \right)ax} \\
& =\frac{\frac{\left( ax-1 \right)ax}{a}}{\frac{\left( ax-1 \right)ax}{x}} \\
& =\frac{\frac{1}{a}}{\frac{1}{x}} \\
& =\frac{x}{a}
\end{align}$
Thus, the simplified form of the expression $\frac{x-{{a}^{-1}}}{a-{{x}^{-1}}}$ is $\frac{x}{a}$.
