Упр.78 ГДЗ Алимов 10-11 класс (Алгебра)

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

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

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

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

Ответ: $$a$$

$$\frac{b^{\frac{1}{5}}\left(\sqrt[5]{b^4}-\sqrt[5]{b^{-1}}\right)}{b^{\frac{2}{3}}\left(\sqrt[3]{b}-\sqrt[3]{b^{-2}}\right)}$$

$$\sqrt[5]{b^4}=b^{\frac{4}{5}},\qquad \sqrt[5]{b^{-1}}=b^{-\frac{1}{5}},\qquad \sqrt[3]{b}=b^{\frac{1}{3}},\qquad \sqrt[3]{b^{-2}}=b^{-\frac{2}{3}}$$

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

Ответ: $$1$$

$$\frac{a^{\frac{5}{3}}b^{-1}-a^{-\frac{1}{3}}}{\sqrt[3]{a^2}}$$

$$\sqrt[3]{a^2}=a^{\frac{2}{3}}$$

$$\frac{a^{\frac{5}{3}}b^{-1}-a^{-\frac{1}{3}}}{a^{\frac{2}{3}}} = a^{\frac{5}{3}-\frac{2}{3}}b^{-1}-a^{-\frac{1}{3}-\frac{2}{3}} = a b^{-1}-a^{-1}$$

$$ab^{-1}-a^{-1}=\frac{a}{b}-\frac{1}{a}=\frac{a^2-b}{ab}$$

Ответ: $$\frac{a^2-b}{ab}$$

$$\frac{a^{\frac{1}{3}}\sqrt{b}+b^{\frac{1}{3}}\sqrt{a}}{\sqrt[6]{a}+\sqrt[6]{b}}$$

$$a^{\frac{1}{3}}\sqrt{b}=a^{\frac{2}{6}}b^{\frac{3}{6}},\qquad b^{\frac{1}{3}}\sqrt{a}=b^{\frac{2}{6}}a^{\frac{3}{6}}$$

$$a^{\frac{1}{3}}\sqrt{b}+b^{\frac{1}{3}}\sqrt{a} = a^{\frac{2}{6}}b^{\frac{3}{6}}+b^{\frac{2}{6}}a^{\frac{3}{6}} = a^{\frac{2}{6}}b^{\frac{2}{6}}\left(a^{\frac{1}{6}}+b^{\frac{1}{6}}\right)$$

$$\frac{a^{\frac{2}{6}}b^{\frac{2}{6}}\left(a^{\frac{1}{6}}+b^{\frac{1}{6}}\right)}{a^{\frac{1}{6}}+b^{\frac{1}{6}}} = a^{\frac{2}{6}}b^{\frac{2}{6}} = a^{\frac{1}{3}}b^{\frac{1}{3}}$$

Ответ: $$a^{\frac{1}{3}}b^{\frac{1}{3}}$$