To find the product of \( \frac{2}{5} \) and \( \frac{2}{9} \), you multiply the numerators together and the denominators together:
\[ \frac{2}{5} \times \frac{2}{9} = \frac{2 \times 2}{5 \times 9} = \frac{4}{45} \]
Therefore, the product of \( \frac{2}{5} \) and \( \frac{2}{9} \) is \( \frac{4}{45} \).