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$\sqrt[4]{\frac{4}{3}\sqrt[3]{\frac{3}{4}}\sqrt{\frac{4}{3}}}=\sqrt[4]{\frac{4}{3}\sqrt[3]{\frac{3}{4}{(\frac{4}{3})^{\frac{1}{2}}}}}=\sqrt[4]{\frac{4}{3}\sqrt[3]{(\frac{3}{4})^{\frac{1}{2}}}}}=\\\ \\\ =\sqrt[4]{(\frac{4}{3})^{\frac{5}{6}}}=(\frac{4}{3})^{\frac{5}{24}}}$

$\frac{\sqrt[3]9+\sqrt3}{\sqrt[3]3+2\sqrt[6]3+1}=\frac{3^{\frac{2}{3}}+3^{\frac{1}{2}}}{3^{\frac{1}{3}}+2*3^{\frac{1}{6}}+1}}}=\frac{3^{\frac{4}{6}}+3^{\frac{3}{6}}}{3^{\frac{2}{6}}+2*3^{\frac{1}{6}}+1}}}=\\\ \\\ =\frac{3^{\frac{3}{6}}(3^{\frac{1}{6}}+1)}{(3^{\frac{1}{6}})^2+2*3^{\frac{1}{6}}+1}}}=\frac{3^{\frac{1}{2}}(3^{\frac{1}{6}}+1)}{(3^{\frac{1}{6}}+1)^2}=\frac{3^{\frac{1}{2}}}{3^{\frac{1}{6}}+1}=\frac{\sqrt3}{\sqrt[6]3+1}$