\[\left( 2\frac{1}{3} \right)^{7} \cdot \left( \frac{3}{7} \right)^{8} = \left( \frac{7}{3} \right)^{7} \cdot \left( \frac{3}{7} \right)^{8} =\]
\[= \frac{7^{7} \cdot 3^{8}}{3^{7} \cdot 7^{8}} = \frac{3}{7}\]