Найдите значение числового выражения: 35.2. a) $$\sqrt[5]{243 \cdot \frac{1}{32}}$$; б) $$\sqrt[3]{\frac{8}{125}}$$; в) $$\sqrt[6]{64 \cdot \frac{1}{729}}$$; г) $$\sqrt[7]{\frac{19}{32}}$$ 35.3. a) $$\sqrt[3]{24 \cdot 9}$$; б) $$\sqrt[5]{48 \cdot 162}$$; в) $$\sqrt[3]{75 \cdot 45}$$; г) $$\sqrt[4]{54 \cdot 24}$$ 35.4. a) $$\sqrt[4]{\frac{125}{0,2}}$$; б) $$\sqrt[4]{\frac{16}{0,0625}}$$; в) $$\sqrt[3]{\frac{27}{0,125}}$$; г) $$\sqrt[6]{\frac{16}{0,25}}$$ 35.5. a) $$\sqrt[3]{5^6 \cdot 2^9}$$; б) $$\sqrt[5]{0,2^{10} \cdot 10^{10}}$$; в) $$\sqrt[3]{0,2^3 \cdot 5^6}$$; г) $$\sqrt[6]{36^3 \cdot 2^6}$$. 35.6. a) $$\sqrt[4]{\frac{7^8}{3^4}}$$; б) $$\sqrt[3]{\frac{5^6}{3^9}}$$; в) $$\sqrt[4]{\frac{3^{12}}{2^8}}$$; г) $$\sqrt[5]{\frac{5^5}{13^{10}}}$$

Решение:

1. 35.2. a) $$\sqrt[5]{243 \cdot \frac{1}{32}} = \sqrt[5]{\frac{243}{32}} = \sqrt[5]{\frac{3^5}{2^5}} = \frac{3}{2} = $$\textbf{1,5}
2. 35.2. б) $$\sqrt[3]{\frac{8}{125}} = \sqrt[3]{\frac{2^3}{5^3}} = \frac{2}{5} = $$\textbf{0,4}
3. 35.2. в) $$\sqrt[6]{64 \cdot \frac{1}{729}} = \sqrt[6]{\frac{64}{729}} = \sqrt[6]{\frac{2^6}{3^6}} = \frac{2}{3} = $$\textbf{0,(6)}
4. 35.2. г) $$\sqrt[7]{\frac{19}{32}}$$ - невозможно упростить без калькулятора. Оставим в виде $$\sqrt[7]{\frac{19}{32}} \approx $$\textbf{0.942}
5. 35.3. a) $$\sqrt[3]{24 \cdot 9} = \sqrt[3]{8 \cdot 3 \cdot 9} = \sqrt[3]{2^3 \cdot 3^3} = 2 \cdot 3 = $$\textbf{6}
6. 35.3. б) $$\sqrt[5]{48 \cdot 162} = \sqrt[5]{16 \cdot 3 \cdot 81 \cdot 2} = \sqrt[5]{2^4 \cdot 3 \cdot 3^4 \cdot 2} = \sqrt[5]{2^5 \cdot 3^5} = 2 \cdot 3 = $$\textbf{6}
7. 35.3. в) $$\sqrt[3]{75 \cdot 45} = \sqrt[3]{25 \cdot 3 \cdot 15 \cdot 3} = \sqrt[3]{5^2 \cdot 3 \cdot 5 \cdot 3 \cdot 3} = \sqrt[3]{5^3 \cdot 3^3} = 5 \cdot 3 = $$\textbf{15}
8. 35.3. г) $$\sqrt[4]{54 \cdot 24} = \sqrt[4]{27 \cdot 2 \cdot 8 \cdot 3} = \sqrt[4]{3^3 \cdot 2 \cdot 2^3 \cdot 3} = \sqrt[4]{3^4 \cdot 2^4} = 3 \cdot 2 = $$\textbf{6}
9. 35.4. a) $$\sqrt[4]{\frac{125}{0,2}} = \sqrt[4]{\frac{125}{\frac{2}{10}}} = \sqrt[4]{\frac{125}{\frac{1}{5}}} = \sqrt[4]{125 \cdot 5} = \sqrt[4]{625} = \sqrt[4]{5^4} = $$\textbf{5}
10. 35.4. б) $$\sqrt[4]{\frac{16}{0,0625}} = \sqrt[4]{\frac{16}{\frac{625}{10000}}} = \sqrt[4]{\frac{16 \cdot 10000}{625}} = \sqrt[4]{\frac{16 \cdot 16 \cdot 625}{625}} = \sqrt[4]{16 \cdot 16} = \sqrt[4]{2^4 \cdot 2^4} = 2 \cdot 2 = $$\textbf{4}
11. 35.4. в) $$\sqrt[3]{\frac{27}{0,125}} = \sqrt[3]{\frac{27}{\frac{125}{1000}}} = \sqrt[3]{\frac{27 \cdot 1000}{125}} = \sqrt[3]{\frac{27 \cdot 8 \cdot 125}{125}} = \sqrt[3]{27 \cdot 8} = \sqrt[3]{3^3 \cdot 2^3} = 3 \cdot 2 = $$\textbf{6}
12. 35.4. г) $$\sqrt[6]{\frac{16}{0,25}} = \sqrt[6]{\frac{16}{\frac{1}{4}}} = \sqrt[6]{16 \cdot 4} = \sqrt[6]{2^4 \cdot 2^2} = \sqrt[6]{2^6} = $$\textbf{2}
13. 35.5. a) $$\sqrt[3]{5^6 \cdot 2^9} = \sqrt[3]{5^6} \cdot \sqrt[3]{2^9} = 5^{\frac{6}{3}} \cdot 2^{\frac{9}{3}} = 5^2 \cdot 2^3 = 25 \cdot 8 = $$\textbf{200}
14. 35.5. б) $$\sqrt[5]{0,2^{10} \cdot 10^{10}} = \sqrt[5]{(0,2 \cdot 10)^{10}} = \sqrt[5]{2^{10}} = 2^{\frac{10}{5}} = 2^2 = $$\textbf{4}
15. 35.5. в) $$\sqrt[3]{0,2^3 \cdot 5^6} = \sqrt[3]{0,2^3} \cdot \sqrt[3]{5^6} = 0,2 \cdot 5^{\frac{6}{3}} = 0,2 \cdot 5^2 = 0,2 \cdot 25 = $$\textbf{5}
16. 35.5. г) $$\sqrt[6]{36^3 \cdot 2^6} = \sqrt[6]{36^3} \cdot \sqrt[6]{2^6} = \sqrt[6]{(6^2)^3} \cdot 2 = \sqrt[6]{6^6} \cdot 2 = 6 \cdot 2 = $$\textbf{12}
17. 35.6. a) $$\sqrt[4]{\frac{7^8}{3^4}} = \frac{\sqrt[4]{7^8}}{\sqrt[4]{3^4}} = \frac{7^{\frac{8}{4}}}{3^{\frac{4}{4}}} = \frac{7^2}{3} = \frac{49}{3} = 16\frac{1}{3} = $$\textbf{16,(3)}
18. 35.6. б) $$\sqrt[3]{\frac{5^6}{3^9}} = \frac{\sqrt[3]{5^6}}{\sqrt[3]{3^9}} = \frac{5^{\frac{6}{3}}}{3^{\frac{9}{3}}} = \frac{5^2}{3^3} = \frac{25}{27} \approx $$\textbf{0.926}
19. 35.6. в) $$\sqrt[4]{\frac{3^{12}}{2^8}} = \frac{\sqrt[4]{3^{12}}}{\sqrt[4]{2^8}} = \frac{3^{\frac{12}{4}}}{2^{\frac{8}{4}}} = \frac{3^3}{2^2} = \frac{27}{4} = $$\textbf{6,75}
20. 35.6. г) $$\sqrt[5]{\frac{5^5}{13^{10}}} = \frac{\sqrt[5]{5^5}}{\sqrt[5]{13^{10}}} = \frac{5}{13^{\frac{10}{5}}} = \frac{5}{13^2} = \frac{5}{169} \approx $$\textbf{0.0296}