$3^{\frac{8}{7}}=\sqrt[7]{3^8}=\sqrt[7]{3^7\cdot 3}=\sqrt[7]{3^7}\cdot \sqrt[7]{3}=3\sqrt[7]{3}$  

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$7^{\frac{5}{3}}=\sqrt[3]{7^5}=\sqrt[3]{7^3\cdot 7^2}=\sqrt[3]{7^3}\cdot \sqrt[3]{7^2}=7\sqrt[3]{49}$  

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${\left(\frac{5}{4}\right)}^{1,5}={\left(\frac{5}{4}\right)}^{\frac{3}{2}}=\sqrt{{\left(\frac{5}{4}\right)}^3}=\sqrt{{\left(\frac{5}{4}\right)}^2\cdot \left(\frac{5}{4}\right)}=\sqrt{{\left(\frac{5}{4}\right)}^2}\cdot \sqrt{\left(\frac{5}{4}\right)}=\frac{5}{4}\cdot \left(\frac{\sqrt{5}}{\sqrt{4}}\right)=$  

$=\frac{5}{4}\cdot \frac{\sqrt{5}}{2}=\frac{5\sqrt{5}}{8}$  

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${0,1}^{-\frac{8}{5}}=\sqrt[5]{{0,1}^{-8}}=\sqrt[5]{{\left(\frac{1}{10}\right)}^{-8}}=\sqrt[5]{{10}^8}=\sqrt[5]{{10}^5\cdot {10}^3}=\sqrt[5]{{10}^5}\cdot \sqrt[5]{{10}^3}=10\sqrt[5]{{10}^3}=10\sqrt[5]{1000}$  

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${0,9}^{-1,5}={0,9}^{-\frac{3}{2}}=\sqrt{{0,9}^{-3}}=\sqrt{{\left(\frac{9}{10}\right)}^{-3}}=\sqrt{{\left(\frac{10}{9}\right)}^3}=\sqrt{{\left(\frac{10}{9}\right)}^2\cdot \frac{10}{9}}=$  

$=\sqrt{{\left(\frac{10}{9}\right)}^2}\cdot \sqrt{\frac{10}{9}}=\frac{10}{9}\cdot \sqrt{\frac{10}{9}}=\frac{10}{9}\cdot \frac{\sqrt{10}}{\sqrt{9}}=\frac{10}{9}\cdot \frac{\sqrt{10}}{3}=\frac{10\sqrt{10}}{27}$