9). Вычислите: a) 81^(3/4); б) 16^(-0.75); в) 0,0625^(-1/4); г) ³√(-3/8); д) ³√(2⁴/4); е) ³√(√3)/9

Решение:

1. a) \( 81^{\frac{3}{4}} \)
   \( 81^{\frac{3}{4}} = (81^{\frac{1}{4}})^3 = (\sqrt[4]{81})^3 = 3^3 = 27 \).
2. б) \( 16^{-0.75} \)
   \( 0.75 = \frac{3}{4} \>.
   \( 16^{-\frac{3}{4}} = \frac{1}{16^{\frac{3}{4}}} = \frac{1}{(16^{\frac{1}{4}})^3} = \frac{1}{(\sqrt[4]{16})^3} = \frac{1}{2^3} = \frac{1}{8} \).
3. в) \( 0.0625^{-\frac{1}{4}} \)
   \( 0.0625 = \frac{625}{10000} = \frac{1}{16} \>.
   \( (\frac{1}{16})^{-\frac{1}{4}} = 16^{\frac{1}{4}} = \sqrt[4]{16} = 2 \).
4. г) \( \sqrt[3]{-\frac{3}{8}} \)
   \( \sqrt[3]{-\frac{3}{8}} = -\sqrt[3]{\frac{3}{8}} = -\frac{\sqrt[3]{3}}{\sqrt[3]{8}} = -\frac{\sqrt[3]{3}}{2} \).
5. д) \( \sqrt[3]{\frac{2^4}{4}} \)
   \( \sqrt[3]{\frac{2^4}{4}} = \sqrt[3]{\frac{16}{4}} = \sqrt[3]{4} \).
6. е) \( \frac{\sqrt[3]{\sqrt{3}}}{9} \)
   \( \frac{\sqrt[3]{\sqrt{3}}}{9} = \frac{(3^{\frac{1}{2}})^{\frac{1}{3}}}{9} = \frac{3^{\frac{1}{6}}}{9} = \frac{3^{\frac{1}{6}}}{3^2} = 3^{\frac{1}{6} - 2} = 3^{-\frac{11}{6}} \>.

Ответ: a) \( 27 \); б) \( \frac{1}{8} \); в) \( 2 \); г) \( -\frac{\sqrt[3]{3}}{2} \); д) \( \sqrt[3]{4} \); е) \( 3^{-\frac{11}{6}} \).