$\text{a)}\sqrt[4]{2^3}\cdot 8^{-1\frac{1}{3}}\cdot {\left(\sqrt[3]{16}\right)}^{\frac{3}{2}}=$  

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

$=2^{\frac{3}{4}}\cdot 2^{-4}\cdot 2^2=2^{\frac{3}{4}-4+2}=2^{-1\frac{1}{4}}$  

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$\text{b)}{\left(\frac{1}{27}\right)}^{\frac{1}{4}}:{\left(\sqrt[5]{81}\right)}^{-3}\cdot 3^2={\left(3^{-3}\right)}^{\frac{1}{4}}:{\left(3^{\frac{4}{5}}\right)}^{-3}\cdot 3^2=$  

$=3^{-\frac{3}{4}}:3^{-\frac{12}{5}}\cdot 3^2=3^{-\frac{3}{4}-\left(-\frac{12}{5}\right)+2}=$  

$=3^{-\frac{3}{4}+\frac{12}{5}+2}=3^{-\frac{15}{20}+\frac{48}{20}+2}=3^{\frac{33}{20}+2}=3^{1\frac{13}{20}+2}=3^{3\frac{13}{20}}$