Упр.493 ГДЗ Колягин Ткачёва 10 класс (Алгебра)

2) (a1/3 + b1/3) : (2 + корень 3 степени a/b + корень 3 степени b/a);
3)
4)

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

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

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

4. $$\frac{\sqrt a-a^{-\frac12}b}{1-\sqrt{a^{-1}b}}-\frac{\sqrt[3]{a^2}-a^{-\frac13}b}{\sqrt[6]{b}+\;a^{-\frac13}\sqrt b}$$
   $$=\frac{a^{-\frac12}(a-b)}{a^{-\frac12}\left(a^{\frac12}-b^{\frac12}\right)}-\frac{a^{-\frac13}(a-b)}{a^{-\frac13}\left(a^{\frac16}b^{\frac13}+b^{\frac12}\right)}$$
   $$=\frac{a-b}{a^{\frac12}-b^{\frac12}}-\frac{a-b}{a^{\frac12}+b^{\frac12}}$$
   $$=\frac{(a-b)\left(a^{\frac12}+b^{\frac12}\right)-(a-b)\left(a^{\frac12}-b^{\frac12}\right)}{\left(a^{\frac12}-b^{\frac12}\right)\left(a^{\frac12}+b^{\frac12}\right)}$$
   $$=\frac{2b^{\frac12}(a-b)}{a-b}=2\sqrt b.$$

Ответ

1) $$\frac{1}{a}$$; 2) $$\frac{\sqrt[3]{ab}}{\sqrt[3]{a}+\sqrt[3]{b}}$$; 3) $$a+b$$; 4) $$2\sqrt b$$.