\[\left( - \frac{7}{9}ab^{2} \right) \cdot \left( - \frac{3}{14}a^{4}b^{2} \right) =\]
\[= \frac{7}{9} \cdot \frac{3}{14} \cdot a^{5}b^{4} = \frac{1}{6}a^{5}b^{4}.\]