\[\left( 1\frac{7}{9} \right)^{11} \cdot \left( \frac{9}{16} \right)^{12} =\]
\[= \left( \frac{16}{9} \right)^{11} \cdot \left( \frac{9}{16} \right)^{12} =\]
\[= \frac{16^{11} \cdot 9^{12}}{9^{11} \cdot 16^{12}} = \frac{9}{16}\]