$\begin{align}  & \frac{{{25}^{\tfrac{2}{3}}}\div {{25}^{\tfrac{1}{6}}}}{{{({1}/{5}\;)}^{-\tfrac{7}{6}}}\times {{({1}/{5}\;)}^{\tfrac{1}{6}}}}=\frac{{{25}^{\tfrac{2}{3}-\tfrac{1}{6}}}}{{{(\tfrac{1}{5})}^{-\tfrac{7}{6}+\tfrac{1}{6}}}}=\frac{{{25}^{\tfrac{3}{6}}}}{{{(\tfrac{1}{5})}^{-\tfrac{6}{6}}}} \\ & \frac{{{25}^{\tfrac{2}{3}}}\div {{25}^{\tfrac{1}{6}}}}{{{({1}/{5}\;)}^{-\tfrac{7}{6}}}\times {{({1}/{5}\;)}^{\tfrac{1}{6}}}}=\frac{{{25}^{\tfrac{1}{2}}}}{{{(\tfrac{1}{5})}^{-1}}}=\frac{{{({{5}^{2}})}^{\tfrac{1}{2}}}}{{{({{5}^{-1}})}^{-1}}} \\ & \frac{{{25}^{\tfrac{2}{3}}}\div {{25}^{\tfrac{1}{6}}}}{{{({1}/{5}\;)}^{-\tfrac{7}{6}}}\times {{({1}/{5}\;)}^{\tfrac{1}{6}}}}=\frac{5}{5}=1 \\\end{align}$