1. Вычислить: a) (7^1/3 * 7^-2/3) / 7^-3 б) (cbrt(sqrt(8)))^2 в) (20.5)^-0.5 * (0.5)^-1.25 г) cbrt(50) * cbrt(20)

1. Вычисление:

1. а)

   \( \frac{7^{\frac{1}{3}} \cdot 7^{-\frac{2}{3}}}{7^{-3}} = \frac{7^{\frac{1}{3} - \frac{2}{3}}}{7^{-3}} = \frac{7^{-\frac{1}{3}}}{7^{-3}} = 7^{-\frac{1}{3} - (-3)} = 7^{-\frac{1}{3} + 3} = 7^{\frac{8}{3}} \)

2. б)

   \( (\sqrt[3]{\sqrt{8}})^2 = (\sqrt[6]{8})^2 = (8^{\frac{1}{6}})^2 = 8^{\frac{2}{6}} = 8^{\frac{1}{3}} = \sqrt[3]{8} = 2 \)

3. в)

   \( (20.5)^{-0.5} \cdot (0.5)^{-1.25} = \left(\frac{41}{2}\right)^{-\frac{1}{2}} \cdot \left(\frac{1}{2}\right)^{-\frac{5}{4}} = \left(\frac{2}{41}\right)^{\frac{1}{2}} \cdot \left(2\right)^{\frac{5}{4}} = \frac{\sqrt{2}}{\sqrt{41}} \cdot 2^{\frac{5}{4}} = \frac{2^{\frac{1}{2}}}{41^{\frac{1}{2}}} \cdot 2^{\frac{5}{4}} = \frac{2^{\frac{1}{2} + \frac{5}{4}}}{41^{\frac{1}{2}}} = \frac{2^{\frac{7}{4}}}{\sqrt{41}} \)

4. г)

   \( \sqrt[3]{50} \cdot \sqrt[3]{20} = \sqrt[3]{50 \cdot 20} = \sqrt[3]{1000} = 10 \)

Ответ: а) 7^8/3; б) 2; в) \(\frac{2^{7/4}}{\sqrt{41}}\); г) 10.