Упростите выражение (1/a^2+1/b^2-(2a-2b)/ab*1/(a-b))*a^2b^2/(a^2-b^2).

Ответ:

\[\left( \frac{1}{a^{2}} + \frac{1}{b^{2}} - \frac{2a - 2b}{\text{ab}} \cdot \frac{1}{a - b} \right) \cdot \frac{a^{2}b^{2}\ }{a^{2} - b^{2}} =\]

\[= \frac{a - b}{a + b}\]

\[1)\ \frac{2a - 2b}{\text{ab}} \cdot \frac{1}{a - b} = \frac{2 \cdot (a - b)}{\text{ab}(a - b)} = \frac{2}{\text{ab}}\]

\[2)\ \frac{1^{\backslash b^{2}}}{a^{2}} + \frac{1^{\backslash a^{2}}}{b^{2}} - \frac{2^{\backslash ab}}{\text{ab}} = \frac{b^{2} + a^{2} - 2ab}{a^{2}b^{2}} =\]

\[= \frac{(a - b)^{2}}{a^{2}b^{2}}\]

\[3)\ \frac{(a - b)^{2}}{a^{2}b^{2}} \cdot \frac{a^{2}b^{2}\ }{a^{2} - b^{2}} = \frac{(a - b)(a - b)}{(a - b)(a + b)} =\]

\[= \frac{a - b}{a + b}\ \]