Question

Помогите с решением пожалуйста №2

Answer 1

№2.
1)
$frac{y^2(xy^{-1}-1)^2}{x(1+x^{-1}y)^2}* frac{y^2(x^{-2}+y^{-2})}{x(xy^{-1}+x^{-1}y)}= frac{y^2( frac{x}{y} -1)^2}{x(1+ frac{y}{x} )^2}* frac{y^2( frac{1}{x^2}+ frac{1}{y^2} )}{x*( frac{x}{y} + frac{y}{x} )}= \ \ = frac{y^2( frac{x-y}{y} )^2}{x( frac{x+y}{x} )^2} * frac{y^2( frac{x^2+y^2}{x^2y^2} )}{x( frac{x^2+y^2}{xy} )}= frac{y^2* frac{(x-y)^2}{y^2} }{x* frac{(x+y)^2}{x^2} } * frac{ frac{x^2+y^2}{x^2} }{ frac{x^2+y^2}{y} }=$
$= frac{(x-y)^2}{ frac{(x+y)^2}{x} } *frac{x^2+y^2}{x^2}* frac{y}{x^2+y^2}= frac{x(x-y)^2}{(x+y)^2}* frac{y}{x^2}= frac{y(x-y)^2}{x(x+y)^2}$

2)
$frac{1-x^{-1}y}{xy^{-1}+1}= frac{1- frac{y}{x} }{ frac{x}{y} +1}= frac{ frac{x-y}{x} }{ frac{x+y}{y} }= frac{x-y}{x}* frac{y}{x+y}= frac{y(x-y)}{x(x+y)} }$

3)
$frac{y(x-y)^2}{x(x+y)^2}: frac{y(x-y)}{x(x+y)}= frac{y(x-y)^2}{x(x+y)^2}* frac{x(x+y)}{y(x-y)}= frac{x-y}{x+y}$