1. **State the problem:** Multiply the fractions $\frac{4}{7}$ and $\frac{8}{9}$. 2. **Formula:** To multiply fractions, multiply the numerators together and the denominators together: $$\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$$ 3. **Apply the formula:** $$\frac{4}{7} \times \frac{8}{9} = \frac{4 \times 8}{7 \times 9} = \frac{32}{63}$$ 4. **Simplify if possible:** Check if numerator and denominator have common factors. Since 32 and 63 share no common factors other than 1, the fraction is already in simplest form. 5. **Final answer:** $$\frac{4}{7} \times \frac{8}{9} = \frac{32}{63}$$