Упр.81 ГДЗ Алимов 10-11 класс (Алгебра)

1)

$$\left(1-2\sqrt{\frac{b}{a}}+\frac{b}{a}\right):\left(a^{\frac12}-b^{\frac12}\right)^2 = \left(1-\sqrt{\frac{b}{a}}\right)^2:\left(a^{\frac12}-b^{\frac12}\right)^2$$
$$= \left(\frac{a^{\frac12}-b^{\frac12}}{a^{\frac12}}\right)^2:\left(a^{\frac12}-b^{\frac12}\right)^2 = \frac{1}{a}$$

Ответ: $$\frac{1}{a}$$

2)

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

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

3)

$$\frac{a^{\frac14}-a^{\frac94}}{a^{\frac14}-a^{\frac54}} — \frac{b^{-\frac12}-b^{\frac32}}{b^{-\frac12}+b^{-\frac12}} = \frac{a^{\frac14}(1-a^2)}{a^{\frac14}(1-a)} — \frac{b^{-\frac12}(1-b^2)}{b^{-\frac12}(1+b)}$$
$$= \frac{1-a^2}{1-a}-\frac{1-b^2}{1+b} = \frac{(1-a)(1+a)}{1-a}-\frac{(1-b)(1+b)}{1+b}$$
$$= (1+a)-(1-b)=a+b$$

Ответ: $$a+b$$

4)

$$\frac{\sqrt{a}-a^{-\frac12}b}{1-\sqrt{a^{-1}b}} \cdot \frac{\sqrt[3]{a^2}-a^{-\frac13}b}{\sqrt[6]{a}+a^{-\frac13}\sqrt{b}} = \frac{a^{\frac12}-a^{-\frac12}b}{1-a^{-\frac12}b^{\frac12}} \cdot \frac{a^{\frac23}-a^{-\frac13}b}{a^{\frac16}+a^{-\frac13}b^{\frac12}}$$
$$= \frac{a^{-\frac12}(a-b)}{a^{-\frac12}(a^{\frac12}-b^{\frac12})} \cdot \frac{a^{-\frac13}(a-b)}{a^{-\frac13}(a^{\frac12}+b^{\frac12})} = \frac{a-b}{a^{\frac12}-b^{\frac12}} \cdot \frac{a-b}{a^{\frac12}+b^{\frac12}}$$
$$= \frac{(a-b)^2}{a-b} = a-b$$

Ответ: $$a-b$$