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

1)

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

Ответ: $$2\sqrt[3]{ab}$$

2)

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

Ответ: $$2\sqrt[3]{b}$$

3)

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

Ответ: $$-\frac{\sqrt[3]{ab}}{a-b}$$

4)

$$\frac{a^{2/3}+b^{2/3}}{a+b}+\frac{1}{a^{2/3}-a^{1/3}b^{1/3}+b^{2/3}}$$
$$= \frac{a^{1/3}-b^{1/3}}{a-b}+\frac{a^{1/3}+b^{1/3}}{a-b} = \frac{2a^{1/3}}{a-b}.$$

Ответ: $$\frac{2\sqrt[3]{a}}{a-b}$$