Упр.88 ГДЗ Алимов 10-11 класс (Алгебра)

1)

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

Ответ: $$2\sqrt[3]{ab}$$

2)

$$\frac{a+b}{a^{\frac23}-a^{\frac13}b^{\frac13}+b^{\frac23}}-\frac{a-b}{a^{\frac23}+a^{\frac13}b^{\frac13}+b^{\frac23}}$$
$$= \frac{(a+b)\left(a^{\frac13}+b^{\frac13}\right)}{a+b} — \frac{(a-b)\left(a^{\frac13}-b^{\frac13}\right)}{a-b}$$
$$= \left(a^{\frac13}+b^{\frac13}\right)-\left(a^{\frac13}-b^{\frac13}\right) = 2b^{\frac13}.$$

Ответ: $$2\sqrt[3]{b}$$

3)

$$\frac{a^{\frac23}+b^{\frac23}}{a-b}-\frac{1}{a^{\frac13}-b^{\frac13}} = \frac{a^{\frac23}+b^{\frac23}}{a-b} — \frac{a^{\frac23}+a^{\frac13}b^{\frac13}+b^{\frac23}}{(a^{\frac13}-b^{\frac13})(a^{\frac23}+a^{\frac13}b^{\frac13}+b^{\frac23})}$$
$$= \frac{a^{\frac23}+b^{\frac23}-(a^{\frac23}+a^{\frac13}b^{\frac13}+b^{\frac23})}{a-b} = -\frac{a^{\frac13}b^{\frac13}}{a-b} = -\frac{\sqrt[3]{ab}}{a-b}.$$

Ответ: $$-\frac{\sqrt[3]{ab}}{a-b}$$

4)

$$\frac{a^{\frac23}+b^{\frac23}}{a+b}+\frac{1}{a^{\frac23}-a^{\frac13}b^{\frac13}+b^{\frac23}}$$
$$= \frac{a^{\frac13}-b^{\frac13}}{a+b}+\frac{a^{\frac13}+b^{\frac13}}{a+b} = \frac{2a^{\frac13}}{a+b} = \frac{2\sqrt[3]{a}}{a+b}.$$

Ответ: $$\frac{2\sqrt[3]{a}}{a+b}$$