N° 10 Espressioni Algebriche

$[{(x - \frac{x^{2}}{x + y})}^{2} : {(y + \frac{y^{2}}{x - y})}^{2}] : (\frac{x}{y} - \frac{y}{x}) : (\frac{x}{x + y} + \frac{y}{x - y}) \cdot (\frac{x + y}{x - y} + \frac{x - y}{x + y}) : {(\frac{xy}{x + y})}^{2} : \frac{xy (x + y)}{x - y} =$

$= [{(\frac{x^{2} + xy - x^{2}}{x + y})}^{2} : {(\frac{xy - y^{2} + y^{2}}{x - y})}^{2}] : (\frac{x^{2} - y^{2}}{xy}) \cdot (\frac{x^{2} - xy + xy + y^{2}}{(x + y) (x - y)}) : (\frac{x^{2} + 2 xy + y^{2} + x^{2} - 2 xy + y^{2}}{(x + y) (x - y)}) : \frac{x^{2} y^{2}}{{(x + y)}^{2}} \cdot \frac{(x - y)}{xy (x + y)} =$

$= [\frac{x^{2} y^{2}}{{(x + y)}^{2}} : \frac{x^{2} y^{2}}{{(x - y)}^{2}}] : \frac{(x + y) (x - y)}{xy} \cdot (\frac{x^{2} + y^{2}}{(x + y) (x - y)}) : (\frac{2 (x^{2} + y^{2})}{(x + y) (x - y)}) \cdot \frac{{(x + y)}^{3}}{xy (x - y)} =$

$= [\frac{x^{2} y^{2}}{{(x + y)}^{2}} \cdot \frac{{(x - y)}^{2}}{x^{2} y^{2}}] : \frac{(x + y) (x - y)}{xy} \cdot \frac{x^{2} + y^{2}}{(x + y) (x - y)} : \frac{2 (x^{2} + y^{2}) {(x + y)}^{2}}{xy {(x - y)}^{2}} =$

$= \frac{{(x - y)}^{2}}{{(x + y)}^{2}} : \frac{(x + y) (x - y)}{xy} \cdot \frac{x^{2} + y^{2}}{(x + y) (x - y)} \cdot \frac{xy {(x - y)}^{2}}{2 (x^{2} + y^{2}) {(x + y)}^{2}} =$

$= \frac{{(x - y)}^{2}}{{(x + y)}^{2}} \cdot \frac{{2 (x + y)}^{2}}{{(x - y)}^{2}} = 2$