Упростите выражение (1/x^2+1/y^2+1/(x+y)*(2x+2y)/xy)*x^2y^2/(x^2-y^2).

Ответ:

\[\left( \frac{1}{x^{2}} + \frac{1}{y^{2}} + \frac{1}{x + y} \cdot \frac{2x + 2y}{\text{xy}} \right) \cdot \frac{x^{2}y^{2}}{x^{2} - y^{2}} =\]

\[= \frac{x + y}{x - y}\]

\[1)\ \frac{1}{x + y} \cdot \frac{2x + 2y}{\text{xy}} = \frac{2 \cdot (x + y)}{(x + y) \cdot xy} = \frac{2}{\text{xy}}\]

\[2)\ \frac{1^{\backslash y^{2}}}{x^{2}} + \frac{1^{\backslash x^{2}}}{y^{2}} + \frac{2^{\backslash xy}}{\text{xy}} = \frac{x^{2} + 2xy + y^{2}}{x^{2}y^{2}} =\]

\[= \frac{(x + y)^{2}}{x^{2}y^{2}}\]

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

\[= \frac{x + y}{x - y}\ \]