$\begin{align}  & {{\left( \frac{16}{81} \right)}^{\tfrac{1}{4}}}\div {{\left( \frac{9}{16} \right)}^{-\tfrac{1}{2}}}={{\left( \frac{16}{81} \right)}^{\tfrac{1}{4}}}\times {{\left( \frac{16}{9} \right)}^{-\tfrac{1}{2}}}={{\left( \frac{{{2}^{4}}}{{{3}^{4}}} \right)}^{\tfrac{1}{4}}}\times {{\left( \frac{{{4}^{2}}}{{{3}^{2}}} \right)}^{-\frac{1}{2}}} \\ & {{\left( \frac{16}{81} \right)}^{\tfrac{1}{4}}}\div {{\left( \frac{9}{16} \right)}^{-\tfrac{1}{2}}}={{\left[ {{\left( \frac{2}{3} \right)}^{4}} \right]}^{\tfrac{1}{4}}}\times {{\left[ {{\left( \frac{4}{3} \right)}^{2}} \right]}^{-\tfrac{1}{2}}}=\frac{2}{3}\times \frac{3}{4}=\frac{1}{2} \\\end{align}$