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

Рассмотрим вариант решения задания из учебника Алимов, Колягин, Ткачёва 10 класс, Просвещение:

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

2. $$\left(a^{\frac13}-b^{\frac13}\right):\left(2+\sqrt[3]{\frac{a}{b}}+\sqrt[3]{\frac{b}{a}}\right)$$
   $$=\left(a^{\frac13}-b^{\frac13}\right):\left(2+\frac{a^{\frac13}}{b^{\frac13}}+\frac{b^{\frac13}}{a^{\frac13}}\right)$$
   $$=\left(a^{\frac13}-b^{\frac13}\right):\frac{a^{\frac13}b^{\frac13}+a^{\frac23}+b^{\frac23}}{a^{\frac13}b^{\frac13}}$$
   $$=\left(a^{\frac13}-b^{\frac13}\right)\cdot \frac{a^{\frac13}b^{\frac13}}{a^{\frac13}b^{\frac13}+a^{\frac23}+b^{\frac23}}.$$
   Так как
   $$a^{\frac13}b^{\frac13}+a^{\frac23}+b^{\frac23}=\left(a^{\frac13}+b^{\frac13}\right)^2-a^{\frac13}b^{\frac13},$$
   то удобнее записать выражение в виде
   $$\frac{a^{\frac13}b^{\frac13}}{a^{\frac13}+b^{\frac13}}.$$

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

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

Ответ

1) $$\frac{1}{a}$$; 2) $$\frac{a^{\frac13}b^{\frac13}}{a^{\frac13}+b^{\frac13}}$$; 3) $$a+b$$; 4) $$2\sqrt{b}$$.