Упр.68 ГДЗ Колягин Ткачёва 9 класс (Алгебра)

1) (a^(4/3)(a^(-1/3)+a^(2/3)))/(a^(1/4)(a^(3/4)+a^(-1/4)));
2) (b^(1/5)((b^4)^(1/5)-(b^(-1))^(1/5)))/(b^(2/3)(b^(1/3)-(b^(-2))^(1/3)));
3) (a^(5/3)b^(-1)-ab^(-1/3))/((a^2)^(1/3)-(b^2)^(1/3));
4) (a^(1/3)vb+b^(1/3)va)/(a^(1/6)+b^(1/6)).

1. $$\frac{a^{4/3}\left(a^{-1/3}+a^{2/3}\right)}{a^{1/4}\left(a^{3/4}+a^{-1/4}\right)}= \frac{a^{4/3}\cdot a^{-1/3}+a^{4/3}\cdot a^{2/3}}{a^{1/4}\cdot a^{3/4}+a^{1/4}\cdot a^{-1/4}}= \frac{a+a^2}{a+1}$$

   $$\frac{a+a^2}{a+1}=\frac{a(1+a)}{a+1}=a$$

2. $$\frac{b^{1/5}\left((b^4)^{1/5}-(b^{-1})^{1/5}\right)}{b^{2/3}\left(b^{1/3}-(b^{-2})^{1/3}\right)}= \frac{b^{1/5}\left(b^{4/5}-b^{-1/5}\right)}{b^{2/3}\left(b^{1/3}-b^{-2/3}\right)}$$

   $$=\frac{b^{1}-b^{0}}{b^{1}-b^{0}}=1$$

3. $$\frac{a^{5/3}b^{-1}-ab^{-1/3}}{(a^2)^{1/3}-(b^2)^{1/3}}= \frac{ab^{-1}\left(a^{2/3}-b^{2/3}\right)}{a^{2/3}-b^{2/3}}=ab^{-1}=\frac{a}{b}$$

4. $$\frac{a^{1/3}\sqrt{b}+b^{1/3}\sqrt{a}}{\sqrt[6]{a}+\sqrt[6]{b}}= \frac{a^{1/3}b^{1/2}+b^{1/3}a^{1/2}}{a^{1/6}+b^{1/6}}$$

   $$=\frac{a^{1/3}b^{1/3}\left(b^{1/6}+a^{1/6}\right)}{a^{1/6}+b^{1/6}}=a^{1/3}b^{1/3}=\sqrt[3]{ab}$$

Ответ

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