Упр.115 ГДЗ Алимов 10-11 класс (Алгебра)

1)

$$\left(\frac{a^{\frac43}b+ab^{\frac43}}{\frac1{a^{\frac13}+b^{\frac13}}\cdot \frac1{a^{\frac13}b^{\frac13}}}\right)^3 = \left(\frac{ab\left(a^{\frac13}+b^{\frac13}\right)}{\frac1{a^{\frac13}+b^{\frac13}}\cdot \frac1{a^{\frac13}b^{\frac13}}}\right)^3 = \left(\frac{ab}{\frac1{a^{\frac13}b^{\frac13}}}\right)^3$$
$$= \left(a^{1-\frac13}b^{1-\frac13}\right)^3 = \left(a^{\frac23}b^{\frac23}\right)^3 = a^2b^2.$$

Ответ: $$a^2b^2$$

2)

$$\frac{a^{\frac13}-b^{\frac13}}{\sqrt[3]{ab}}\cdot \frac{ab^{\frac13}-a^{\frac13}b}{\sqrt[3]{a}+\sqrt[3]{b}} = \frac{a^{\frac13}-b^{\frac13}}{a^{\frac13}b^{\frac13}}\cdot \frac{a^{\frac13}b^{\frac13}\left(a^{\frac23}-b^{\frac23}\right)}{a^{\frac13}+b^{\frac13}}$$
$$= \frac{\left(a^{\frac13}-b^{\frac13}\right)\left(a^{\frac13}-b^{\frac13}\right)\left(a^{\frac13}+b^{\frac13}\right)}{a^{\frac13}+b^{\frac13}} = \left(a^{\frac13}-b^{\frac13}\right)^2 = \left(\sqrt[3]{a}-\sqrt[3]{b}\right)^2.$$

Ответ: $$\left(\sqrt[3]{a}-\sqrt[3]{b}\right)^2$$

3)

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

Ответ: $$1$$

4)

$$\frac{a^{\frac43}-b^{\frac43}}{a^{\frac13}+b^{\frac13}}\cdot \frac{a^{\frac23}+\sqrt[3]{ab}+b^{\frac23}}{a-b} = \frac{\left(a^{\frac23}-b^{\frac23}\right)\left(a^{\frac23}+b^{\frac23}\right)}{a^{\frac13}+b^{\frac13}}\cdot \frac{a^{\frac23}+\sqrt[3]{ab}+b^{\frac23}}{a-b}$$
$$= \frac{\left(a^{\frac23}-b^{\frac23}\right)\left(\left(a^{\frac23}\right)^2-a^{\frac23}b^{\frac23}+\left(b^{\frac23}\right)^2\right)}{a^{\frac13}+b^{\frac13}} = a^2+b^2.$$

Ответ: $$a^2+b^2$$