\[\left( 2\frac{2}{3} \right)^{5} \cdot \left( \frac{3}{8} \right)^{6} = \left( \frac{8}{3} \right)^{5} \cdot \left( \frac{3}{8} \right)^{6} = \frac{8^{5} \cdot 3^{6}}{3^{5} \cdot 8^{6}} = \frac{3}{8}\]