Упростите выражение: (a/(a-c)+2ac/(a^2-2ac+c^2))(4ac/(a+c)-a-c).

Ответ:

\[\left( \frac{a}{a - c} + \frac{2ac}{a^{2} - 2ac + c^{2}} \right)\left( \frac{4ac}{a + c} - a - c \right) =\]

\[1)\ \frac{a}{a - c} + \frac{2ac}{a^{2} - 2ac + c^{2}} = \frac{a^{\backslash a - c}}{a - c} +\]

\[+ \frac{2ac}{(a - c)^{2}} = \frac{a^{2} - ac + 2ac}{(a - c)^{2}} = \frac{a^{2} + ac}{(a - c)^{2}} =\]

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

\[2)\ \frac{4ac}{a + c} - (a + c)^{\backslash a + c} =\]

\[= \frac{4ac - a^{2} - 2ac - c^{2}}{a + c} =\]

\[= \frac{- \left( a^{2} - 2ac + c^{2} \right)}{a + c} = - \frac{(a - c)^{2}}{a + c}\]

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