Answer
The required solution is $\frac{x-y+1}{{{\left( x-y \right)}^{2}}}$
Work Step by Step
We have the given rational expression:
${{\left( x-y \right)}^{-1}}+{{\left( x-y \right)}^{-2}}$
Solve the first bracket of the given expression:
$\begin{align}
& {{\left( x-y \right)}^{-1}}=\frac{1}{{{\left( x-y \right)}^{1}}} \\
& =\frac{1}{x-y}
\end{align}$
Also, solve the second bracket of the given expression:
$\begin{align}
& {{\left( x-y \right)}^{-2}}=\frac{1}{{{\left( x-y \right)}^{2}}} \\
& =\frac{1}{\left( x-y \right)\left( x-y \right)}
\end{align}$
And simplify the given rational expression:
$\begin{align}
& {{\left( x-y \right)}^{-1}}+{{\left( x-y \right)}^{-2}}=\frac{1}{x-y}+\frac{1}{\left( x-y \right)\left( x-y \right)} \\
& =\frac{1}{x-y}\times \frac{\left( x-y \right)}{\left( x-y \right)}+\frac{1}{\left( x-y \right)\left( x-y \right)} \\
& =\frac{x-y}{\left( x-y \right)\left( x-y \right)}+\frac{1}{\left( x-y \right)\left( x-y \right)} \\
& =\frac{x-y+1}{\left( x-y \right)\left( x-y \right)}
\end{align}$
Hence, ${{\left( x-y \right)}^{-1}}+{{\left( x-y \right)}^{-2}}=$ $\frac{x-y+1}{{{\left( x-y \right)}^{2}}}$.
