Упр.37.33 ГДЗ Мордкович 10-11 класс (Алгебра)

Задача

а) $$\frac{a-b}{\sqrt{a}-\sqrt{b}}-\frac{a^{\frac{3}{2}}-b^{\frac{3}{2}}}{a-b}$$; б) $$\frac{\sqrt{x}}{\sqrt{x}+\sqrt{y}}+\frac{\sqrt{y}}{\sqrt{x}-\sqrt{y}}$$.

Подробный ответ

а) $$\frac{a-b}{a^{1/2}-b^{1/2}}-\frac{a^{3/2}-b^{3/2}}{a-b}$$

Представим разность степеней в виде произведения:

$$a-b=\left(a^{1/2}-b^{1/2}\right)\left(a^{1/2}+b^{1/2}\right),$$

$$a^{3/2}-b^{3/2}=\left(a^{1/2}-b^{1/2}\right)\left(a+b^{1/2}a^{1/2}+b\right).$$

Тогда

$$\frac{a-b}{a^{1/2}-b^{1/2}}-\frac{a^{3/2}-b^{3/2}}{a-b} = \frac{\left(a^{1/2}-b^{1/2}\right)\left(a^{1/2}+b^{1/2}\right)}{a^{1/2}-b^{1/2}} — \frac{\left(a^{1/2}-b^{1/2}\right)\left(a+b^{1/2}a^{1/2}+b\right)}{\left(a^{1/2}-b^{1/2}\right)\left(a^{1/2}+b^{1/2}\right)}$$

$$= \left(a^{1/2}+b^{1/2}\right) — \frac{a+a^{1/2}b^{1/2}+b}{a^{1/2}+b^{1/2}} = \frac{\left(a^{1/2}+b^{1/2}\right)^2-\left(a+a^{1/2}b^{1/2}+b\right)}{a^{1/2}+b^{1/2}}$$

$$= \frac{a+2a^{1/2}b^{1/2}+b-a-a^{1/2}b^{1/2}-b}{a^{1/2}+b^{1/2}} = \frac{a^{1/2}b^{1/2}}{a^{1/2}+b^{1/2}} = \frac{\sqrt{ab}}{\sqrt{a}+\sqrt{b}}.$$

б) $$\frac{\sqrt{x}}{x^{1/2}+y^{1/2}}+\frac{\sqrt{y}}{x^{1/2}-y^{1/2}}$$

Так как $$\sqrt{x}=x^{1/2}$$ и $$\sqrt{y}=y^{1/2},$$ получаем

$$\frac{x^{1/2}}{x^{1/2}+y^{1/2}}+\frac{y^{1/2}}{x^{1/2}-y^{1/2}} = \frac{x^{1/2}\left(x^{1/2}-y^{1/2}\right)+y^{1/2}\left(x^{1/2}+y^{1/2}\right)}{\left(x^{1/2}+y^{1/2}\right)\left(x^{1/2}-y^{1/2}\right)}$$

$$= \frac{x-x^{1/2}y^{1/2}+x^{1/2}y^{1/2}+y}{x-y} = \frac{x+y}{x-y}.$$