Упр.3.8 ГДЗ Дорофеев Суворова 9 класс (Алгебра)

а) a + b; б) a — b; в) ab; г) a/b; д) ab/(a + b); е) (a — b)/ab.

Подставим $$a=\frac{xy}{x+y}$$ и $$b=\frac{xy}{x-y}$$ и упростим выражения.

1. $$ a+b=\frac{xy}{x+y}+\frac{xy}{x-y} =\frac{xy(x-y)+xy(x+y)}{(x+y)(x-y)} =\frac{x^2y-xy^2+x^2y+xy^2}{x^2-y^2} =\frac{2x^2y}{x^2-y^2}. $$

2. $$ a-b=\frac{xy}{x+y}-\frac{xy}{x-y} =\frac{xy(x-y)-xy(x+y)}{(x+y)(x-y)} =\frac{x^2y-xy^2-x^2y-xy^2}{x^2-y^2} =\frac{-2xy^2}{x^2-y^2} =\frac{2xy^2}{y^2-x^2}. $$

3. $$ ab=\frac{xy}{x+y}\cdot\frac{xy}{x-y} =\frac{x^2y^2}{x^2-y^2}. $$

4. $$ \frac{a}{b}=\frac{xy}{x+y}:\frac{xy}{x-y} =\frac{xy}{x+y}\cdot\frac{x-y}{xy} =\frac{x-y}{x+y}. $$

5. $$ \frac{ab}{a+b} =\frac{\frac{x^2y^2}{x^2-y^2}}{\frac{2x^2y}{x^2-y^2}} =\frac{x^2y^2}{x^2-y^2}\cdot\frac{x^2-y^2}{2x^2y} =\frac{y}{2}. $$

6. $$ \frac{a-b}{ab} =\frac{\frac{-2xy^2}{x^2-y^2}}{\frac{x^2y^2}{x^2-y^2}} =\frac{-2xy^2}{x^2-y^2}\cdot\frac{x^2-y^2}{x^2y^2} =-\frac{2}{x}. $$

Ответ

а) $$\frac{2x^2y}{x^2-y^2}$$; б) $$\frac{2xy^2}{y^2-x^2}$$; в) $$\frac{x^2y^2}{x^2-y^2}$$; г) $$\frac{x-y}{x+y}$$; д) $$\frac{y}{2}$$; е) $$-\frac{2}{x}$$.