Задача 22. Упростите выражение    \(\left( {\left( {\dfrac{x}{y}-\dfrac{y}{x}} \right)\,{{\left( {\dfrac{x}{y} + \dfrac{y}{x}-2} \right)}^{-1}}} \right)\,\,{\left( {\left( {1 + \dfrac{y}{x}} \right)\dfrac{{{x^{}}}}{{x-y}}} \right)^{-1}}\)

Ответ

ОТВЕТ: 1.

Решение

\(\left( {\left( {\dfrac{x}{y}-\dfrac{y}{x}} \right)\,{{\left( {\dfrac{x}{y} + \dfrac{y}{x}-2} \right)}^{-1}}} \right)\,\,{\left( {\left( {1 + \dfrac{y}{x}} \right)\dfrac{{{x^{}}}}{{x-y}}} \right)^{-1}} = \)

\( = \dfrac{{{x^2}-{y^2}}}{{xy}} \cdot {\left( {\dfrac{{{x^2} + {y^2}-2xy}}{{xy}}} \right)^{-1}} \cdot {\left( {\dfrac{{y + x}}{x} \cdot \dfrac{x}{{x-y}}} \right)^{-1}} = \)

\( = \dfrac{{\left( {x-y} \right)\left( {x + y} \right)}}{{xy}} \cdot \dfrac{{xy}}{{{{\left( {x-y} \right)}^2}}} \cdot \dfrac{x}{{y + x}} \cdot \dfrac{{x-y}}{x} = 1.\)

Ответ:  1.