Answer
$\dfrac{xy^{2}}{y^2+x^2}$
Work Step by Step
The given expression, $
\dfrac{x^{-1}}{x^{-2}+y^{-2}}
,$ simplifies to
\begin{array}{l}\require{cancel}
\dfrac{\dfrac{1}{x^1}}{\dfrac{1}{x^{2}}+\dfrac{1}{y^{2}}}
\\\\=
\dfrac{\dfrac{1}{x}}{\dfrac{y^2+x^2}{x^{2}y^{2}}}
\\\\=
\dfrac{1}{x}\div\dfrac{y^2+x^2}{x^{2}y^{2}}
\\\\=
\dfrac{1}{x}\cdot\dfrac{x^{2}y^{2}}{y^2+x^2}
\\\\=
\dfrac{1}{\cancel{x}}\cdot\dfrac{\cancel{x}\cdot xy^{2}}{y^2+x^2}
\\\\=
\dfrac{xy^{2}}{y^2+x^2}
.\end{array}
