$\text{a)}5\sqrt{5\sqrt{5}}=5\cdot {\left(5\cdot 5^{\frac{1}{2}}\right)}^{\frac{1}{2}}=5\cdot {\left(5^{\frac{3}{2}}\right)}^{\frac{1}{2}}=5\cdot 5^{\frac{3}{4}}=5^{1\frac{3}{4}}=5^{\frac{7}{4}}$  

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$\text{b)}4\sqrt{2\cdot \sqrt[3]{2}}=2^2\cdot {\left(2\cdot 2^{\frac{1}{3}}\right)}^{\frac{1}{2}}=2^2\cdot {\left(2^{\frac{4}{3}}\right)}^{\frac{1}{2}}=2^2\cdot 2^{\frac{2}{3}}=2^{2\frac{2}{3}}=2^{\frac{8}{3}}$  

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$\text{c)}\sqrt[4]{\sqrt[3]{3}}\cdot \sqrt{3\sqrt{3}}={\left(3^{\frac{1}{3}}\right)}^{\frac{1}{4}}\cdot {\left(3\cdot 3^{\frac{1}{2}}\right)}^{\frac{1}{2}}=3^{\frac{1}{12}}\cdot {\left(3^{\frac{3}{2}}\right)}^{\frac{1}{2}}=3^{\frac{1}{12}}\cdot 3^{\frac{3}{4}}=3^{\frac{1}{12}+\frac{9}{12}}=3^{\frac{10}{12}}=3^{\frac{5}{6}}$  

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$\text{d)}36\cdot \sqrt[6]{6\sqrt{6}}=6^2\cdot {\left(6\cdot 6^{\frac{1}{2}}\right)}^{\frac{1}{6}}=6^2\cdot {\left(6^{\frac{3}{2}}\right)}^{\frac{1}{6}}=6^2\cdot 6^{\frac{1}{4}}=6^{2\frac{1}{4}}=6^{\frac{9}{4}}$