\[\left( \frac{7}{9} \right)^{2} - \left( \frac{1}{6} \right)^{2} = \left( \frac{14}{18} \right)^{2} - \left( \frac{3}{18} \right)^{2} =\]
\[= \left( \frac{14}{18} - \frac{3}{18} \right)\left( \frac{14}{18} + \frac{3}{18} \right) =\]
\[= \frac{11}{18} \cdot \frac{17}{18} = \frac{187}{324}.\]