$\text{a)}8\left(x^{\frac{1}{2}}{y}^{\frac{3}{4}}\right)\left(\frac{1}{2}{x}^{\frac{1}{2}}{y}^{\frac{1}{4}}\right)=8\cdot x^{\frac{1}{2}}\cdot y^{\frac{3}{4}}\cdot \frac{1}{2}\cdot x^{\frac{1}{2}}\cdot y^{\frac{1}{4}}=8\cdot \frac{1}{2}\cdot x^{\frac{1}{2}}\cdot x^{\frac{1}{2}}\cdot y^{\frac{3}{4}}\cdot y^{\frac{1}{4}}=$   

$=4\cdot x^{\frac{1}{2}+\frac{1}{2}}\cdot y^{\frac{3}{4}+\frac{1}{4}}=4\cdot x\cdot y=4xy$  

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$\text{b)}\sqrt[3]{a^{-0,5}{b}^{\frac{1}{3}}}=\sqrt[3]{a^{-\frac{1}{2}}{b}^{\frac{1}{3}}}={\left(a^{-\frac{1}{2}}{b}^{\frac{1}{3}}\right)}^{\frac{1}{3}}={\left(a^{-\frac{1}{2}}\right)}^{\frac{1}{3}}\cdot {\left(b^{\frac{1}{3}}\right)}^{\frac{1}{3}}=a^{\left(-\frac{1}{2}\right)\cdot \frac{1}{3}}\cdot b^{\frac{1}{3}\cdot \frac{1}{3}}=$  

$=a^{-\frac{1}{6}}\cdot b^{\frac{1}{9}}=a^{-\frac{1}{6}}{b}^{\frac{1}{9}}$  

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$\text{c)}\sqrt{x\sqrt[3]{y}}:\sqrt[3]{x^{-2}\sqrt{y}}=\frac{\sqrt{x\sqrt[3]{y}}}{\sqrt[3]{x^{-2}\sqrt{y}}}=\frac{\sqrt{x\cdot y^{\frac{1}{3}}}}{\sqrt[3]{x^{-2}\cdot y^{\frac{1}{2}}}}=\frac{{\left(x\cdot y^{\frac{1}{3}}\right)}^{\frac{1}{2}}}{{\left(x^{-2}\cdot y^{\frac{1}{2}}\right)}^{\frac{1}{3}}}=\frac{x^{\frac{1}{2}}\cdot {\left(y^{\frac{1}{3}}\right)}^{\frac{1}{2}}}{{\left(x^{-2}\right)}^{\frac{1}{3}}\cdot {\left(y^{\frac{1}{2}}\right)}^{\frac{1}{3}}}=$  

$=\frac{x^{\frac{1}{2}}\cdot y^{\frac{1}{3}\cdot \frac{1}{2}}}{x^{-2\cdot \frac{1}{3}}\cdot y^{\frac{1}{2}\cdot \frac{1}{3}}}=\frac{x^{\frac{1}{2}}\cdot y^{\frac{1}{6}}}{x^{-\frac{2}{3}}\cdot y^{\frac{1}{6}}}=x^{\frac{1}{2}-\left(-\frac{2}{3}\right)}\cdot y^{\frac{1}{6}-\frac{1}{6}}=x^{\frac{3}{6}+\frac{4}{6}}=x^{\frac{7}{6}}$