Физика
2024
Упростите выражение:
а) \(\left(\frac{2ab}{a^{2}-b^{2}}+\frac{a-b}{2a+2b}\right) \cdot \frac{2a}{a+b}+\frac{b}{b-a}\);
б) \(\frac{y}{x-y}-\frac{x^{3}-xy^{2}}{x^{2}+y^{2}} \cdot \left(\frac{x}{(x-y)^{2}}-\frac{y}{x^{2}-y^{2}}\right)\).
а
\(\left(\frac{2ab}{a^{2}-b^{2}}+\frac{a-b}{2a+2b}\right) \cdot \frac{2a}{a+b}+\frac{b}{b-a}\) =
\(= \left(\frac{2 a b}{(a-b)(a+b)}+\frac{a-b}{2(a+b)}\right) \cdot \frac{2 a}{a+b}-\frac{b}{a-b}\) =
\( = \frac{4 a b+(a-b)^{2}}{2(a-b)(a+b)} \cdot \frac{2 a}{a+b}-\frac{b}{a-b}\) =
\( =\frac{4 a b+a^{2}-2 a b+b^{2}}{2(a-b)(a+b)} \cdot \frac{2 a}{a+b}-\frac{b}{a-b}\) =
\( = \frac{a^{2}+2 a b+b^{2}}{2(a-b)(a+b)} \cdot \frac{2 a}{a+b}-\frac{b}{a-b}\) =
\(= \frac{(a+b)^{2}}{2(a-b)(a+b)} \cdot \frac{2 a}{a+b}-\frac{b}{a-b} = \frac{a}{a-b}-\frac{b}{a-b}\)=
\(= \frac{a-b}{a-b}=1\).
б
\(\frac{y}{x-y}-\frac{x^{3}-x y^{2}}{x^{2}+y^{2}} \cdot\left(\frac{x}{(x-y)^{2}}-\frac{y}{x^{2}-y^{2}}\right)\) =
\(= \frac{y}{x-y}-\frac{x\left(x^{2}-y^{2}\right)}{x^{2}+y^{2}} \cdot\left(\frac{x}{(x-y)^{2}}-\frac{y}{(x-y)(x+y)}\right)\) =
\(= \frac{y}{x-y}-\frac{x\left(x^{2}-y^{2}\right)}{x^{2}+y^{2}} \cdot \frac{x(x+y)-y(x-y)}{(x-y)^{2}(x+y)}\) =
\(= \frac{y}{x-y}-\frac{x}{x^{2}+y^{2}} \cdot \frac{x^{2}+x y-x y+y^{2}}{x-y}\) =
\(= \frac{y}{x-y}-\frac{x}{x^{2}+y^{2}} \cdot \frac{x^{2}+y^{2}}{x-y}=\frac{y}{x-y}-\frac{x}{x-y}\) =
\(= \frac{y-x}{x-y}=-1\).
Решебник
"Алгебра - Учебник" по предмету Математика за 9 класс.
Aвторы:
Макарычев Ю.Н., Миндюк Н.Г., Нешков К.И., Суворова С.Б.
Задание
Упростите выражение: а) \(\left(\frac{2ab}{a^{2}-b^{2}}+\frac{a-b}{2a+2b}\right) \cdot \frac{2a}{a+b}+\frac{b}{b-a}\); б) \(\frac{y}{x-y}-\frac{x^{3}-xy^{2}}{x^{2}+y^{2}} \cdot \left(\frac{x}{(x-y)^{2}}-\frac{y}{x^{2}-y^{2}}\right)\).
