$\text{a)}{\left(3\cdot {27}^{-1}\right)}^{\frac{1}{3}}\cdot {81}^{-\frac{1}{3}}={\left(3\cdot \frac{1}{27}\right)}^{\frac{1}{3}}\cdot {\left(\frac{1}{81}\right)}^{\frac{1}{3}}={\left(\frac{1}{9}\right)}^{\frac{1}{3}}\cdot {\left(\frac{1}{81}\right)}^{\frac{1}{3}}={\left(\frac{1}{9}\cdot \frac{1}{81}\right)}^{\frac{1}{3}}={\left(\frac{1}{729}\right)}^{\frac{1}{3}}=\sqrt[3]{\frac{1}{729}}=\frac{1}{9}$  

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$\text{b)}{\left({25}^{\frac{1}{2}}\cdot 5^{\frac{3}{7}}\right)}^{1,4}={\left(\sqrt{25}\cdot 5^{\frac{3}{7}}\right)}^{1\frac{2}{5}}={\left(5\cdot 5^{\frac{3}{7}}\right)}^{\frac{7}{5}}={\left(5^{1\frac{3}{7}}\right)}^{\frac{7}{5}}={\left(5^{\frac{10}{7}}\right)}^{\frac{7}{5}}=5^2=25$  

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$\text{c)}2\sqrt{8}\cdot 4^{-1,5}\cdot {0,5}^{0,5}=\sqrt{4\cdot 8}\cdot 4^{-\frac{3}{2}}\cdot {\left(\frac{1}{2}\right)}^{\frac{1}{2}}=\sqrt{32}\cdot \sqrt{4^{-3}}\cdot \sqrt{\frac{1}{2}}=\sqrt{32}\cdot \sqrt{\frac{1}{64}}\cdot \sqrt{\frac{1}{2}}=\sqrt{32\cdot \frac{1}{64}\cdot \frac{1}{2}}=\sqrt{\frac{1}{4}}=\frac{1}{2}$  

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$\text{d)}{81}^{0,75}\cdot {16}^{0,25}\cdot {36}^{-0,5}={81}^{\frac{3}{4}}\cdot {16}^{\frac{1}{4}}\cdot \sqrt{\frac{1}{36}}={\left(\sqrt[4]{81}\right)}^3\cdot \sqrt[4]{16}\cdot \frac{1}{6}=3^3\cdot 2\cdot \frac{1}{6}=27\cdot \frac{1}{3}=9$  

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$\text{e)}{0,125}^{-\frac{1}{3}}\cdot {0,04}^{-\frac{3}{2}}\cdot {0,01}^{\frac{1}{2}}={\left(\frac{1}{8}\right)}^{-\frac{1}{3}}\cdot {\left(\sqrt{0,04}\right)}^{-3}\cdot \sqrt{0,01}=8^{\frac{1}{3}}\cdot {0,2}^{-3}\cdot 0,1=\sqrt[3]{8}\cdot {\left(\frac{1}{5}\right)}^{-3}\cdot 0,1=2\cdot 5^3\cdot 0,1=2\cdot 125\cdot 0,1=250\cdot 0,1=25$  

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$\text{f)}{\left[{\left(1\frac{2}{3}\right)}^{\frac{5}{4}}\cdot {0,6}^{-\frac{7}{4}}\cdot {\left(\frac{125}{27}\right)}^{\frac{2}{3}}\right]}^{\frac{1}{5}}={\left[{\left(\frac{5}{3}\right)}^{\frac{5}{4}}\cdot {\left(\frac{3}{5}\right)}^{-\frac{7}{4}}\cdot {\left({\left(\frac{5}{3}\right)}^3\right)}^{\frac{2}{3}}\right]}^{\frac{1}{5}}={\left[{\left(\frac{5}{3}\right)}^{\frac{5}{4}}\cdot {\left(\frac{5}{3}\right)}^{\frac{7}{4}}\cdot {\left(\frac{5}{3}\right)}^2\right]}^{\frac{1}{5}}={\left[{\left(\frac{5}{3}\right)}^{\frac{5}{4}+\frac{7}{4}+2}\right]}^{\frac{1}{5}}={\left[{\left(\frac{5}{3}\right)}^{\frac{12}{4}+2}\right]}^{\frac{1}{5}}={\left[{\left(\frac{5}{3}\right)}^{3+2}\right]}^{\frac{1}{5}}={\left[{\left(\frac{5}{3}\right)}^5\right]}^{\frac{1}{5}}=\frac{5}{3}$