To multiply the fractions \( \frac{5}{9} \) and \( \frac{2}{9} \), you multiply the numerators together and the denominators together:
\[ \frac{5}{9} \times \frac{2}{9} = \frac{5 \times 2}{9 \times 9} = \frac{10}{81} \]
So, \( \frac{5}{9} \times \frac{2}{9} = \frac{10}{81} \).