1. **State the problem:** Multiply the fractions $\frac{2}{3}$ and $\frac{9}{8}$.\n\n2. **Formula:** To multiply fractions, multiply the numerators together and the denominators together: $$\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}.$$\n\n3. **Apply the formula:** $$\frac{2}{3} \times \frac{9}{8} = \frac{2 \times 9}{3 \times 8} = \frac{18}{24}.$$\n\n4. **Simplify the fraction:** Find the greatest common divisor (GCD) of 18 and 24, which is 6. Divide numerator and denominator by 6: $$\frac{\cancel{18}^3}{\cancel{24}^4} = \frac{3}{4}.$$\n\n5. **Final answer:** The product of $\frac{2}{3}$ and $\frac{9}{8}$ simplified is $\frac{3}{4}$.