Докажите тождество: (a^2/(a+5)-a^3/(a^2+10a+25)):(a/(a+5)-a^2/(a^2-25))=(5a-a^2)/(a+5).

Ответ:

\[Упростим\ левую\ часть:\]

\[1)\ \frac{{a^{2}}^{\backslash a + 5}}{a + 5} - \frac{a^{3}}{(a + 5)^{2}} =\]

\[= \frac{a^{3} + 5a^{2} - a^{3}}{(a + 5)^{2}} = \frac{5a^{2}}{(a + 5)^{2}}\]

\[2)\ \frac{a^{\backslash a - 5}}{a + 5} - \frac{a^{2}}{(a - 5)(a + 5)} =\]

\[= \frac{a^{2} - 5a - a^{2}}{a^{2} - 25} = - \frac{5a}{a^{2} - 25}\]

\[3)\ \frac{5a^{2}}{(a + 5)^{2}}\ :\left( - \frac{5a}{a^{2} - 25} \right) =\]

\[= \frac{5a^{2}(a - 5)(a + 5)}{- (a + 5)^{2} \cdot 5a} =\]

\[= - \frac{a(a - 5)}{a + 5} = \frac{5a - a^{2}}{a + 5}\]

\[\frac{5a - a^{2}}{a + 5} = \frac{5a - a^{2}}{a + 5}\]

\[Что\ и\ требовалось\ доказать.\]