Упр.87 ГДЗ Алимов 10-11 класс (Алгебра)

1)

$$\frac{a^{\frac32}}{\sqrt a+\sqrt b}-\frac{ab^{\frac12}}{\sqrt b-\sqrt a}-\frac{2a^2}{a-b} = \frac{a^{\frac32}\left(b^{\frac12}-a^{\frac12}\right)-ab^{\frac12}\left(b^{\frac12}+a^{\frac12}\right)}{b-a} +\frac{2a^2}{a-b}$$
$$= \frac{a^{\frac32}b^{\frac12}-a^2-a^{\frac12}b^{\frac12}a-ab+2a^2}{b-a} = \frac{a^2-ab}{b-a} = \frac{-a(b-a)}{b-a} = -a$$

Ответ: $$-a$$.

2)

$$\frac{3xy-y^2}{x-y}-\frac{y\sqrt y}{\sqrt x-\sqrt y}-\frac{y\sqrt x}{\sqrt x+\sqrt y} = \frac{3xy-y^2}{x-y} -\frac{y\sqrt y(\sqrt x+\sqrt y)+y\sqrt x(\sqrt x-\sqrt y)}{(\sqrt x-\sqrt y)(\sqrt x+\sqrt y)}$$
$$= \frac{3xy-y^2}{x-y} -\frac{y\sqrt{xy}+y^2+xy-y\sqrt{xy}}{x-y} = \frac{3xy-y^2-y^2-xy}{x-y} = \frac{2xy-2y^2}{x-y} = \frac{2y(x-y)}{x-y} = 2y$$

Ответ: $$2y$$.

3)

$$\frac{1}{\sqrt[3]{a}+\sqrt[3]{b}}-\frac{\sqrt[3]{a}+\sqrt[3]{b}}{a^{\frac23}-\sqrt[3]{ab}+b^{\frac23}} = \frac{\left(a^{\frac23}-\sqrt[3]{ab}+b^{\frac23}\right)-\left(\sqrt[3]{a}+\sqrt[3]{b}\right)^2}{\left(\sqrt[3]{a}+\sqrt[3]{b}\right)\left(a^{\frac23}-\sqrt[3]{ab}+b^{\frac23}\right)}$$
$$= \frac{a^{\frac23}-\sqrt[3]{ab}+b^{\frac23}-a^{\frac23}-2\sqrt[3]{ab}-b^{\frac23}}{a+b} = \frac{-3\sqrt[3]{ab}}{a+b}$$

Ответ: $$-\frac{3\sqrt[3]{ab}}{a+b}$$.

4)

$$\frac{\sqrt[3]{a^2}+\sqrt[3]{b^2}}{\sqrt[3]{a}-\sqrt[3]{b}} -\frac{a-b}{a^{\frac23}+\sqrt[3]{ab}+b^{\frac23}} = \frac{(\sqrt[3]{a}-\sqrt[3]{b})(\sqrt[3]{a}+\sqrt[3]{b})}{\sqrt[3]{a}-\sqrt[3]{b}} -\frac{a-b}{a^{\frac23}+\sqrt[3]{ab}+b^{\frac23}}$$
$$= \sqrt[3]{a}+\sqrt[3]{b} -\frac{(\sqrt[3]{a}-\sqrt[3]{b})(a^{\frac23}+\sqrt[3]{ab}+b^{\frac23})}{a-b}$$
$$= \sqrt[3]{a}+\sqrt[3]{b}-\left(\sqrt[3]{a}-\sqrt[3]{b}\right) = 2\sqrt[3]{b}$$

Ответ: $$2\sqrt[3]{b}$$.