1. The problem asks to evaluate $\left(3 \frac{3}{4}\right)^2 \times \left(\frac{15}{4}\right)^3$. 2. First, convert the mixed number $3 \frac{3}{4}$ to an improper fraction: $$3 \frac{3}{4} = \frac{3 \times 4 + 3}{4} = \frac{12 + 3}{4} = \frac{15}{4}$$ 3. Now the expression becomes: $$\left(\frac{15}{4}\right)^2 \times \left(\frac{15}{4}\right)^3$$ 4. Use the rule of exponents for the same base: $a^m \times a^n = a^{m+n}$. 5. So, $$\left(\frac{15}{4}\right)^2 \times \left(\frac{15}{4}\right)^3 = \left(\frac{15}{4}\right)^{2+3} = \left(\frac{15}{4}\right)^5$$ 6. Calculate $\left(\frac{15}{4}\right)^5$: $$\left(\frac{15}{4}\right)^5 = \frac{15^5}{4^5} = \frac{759375}{1024}$$ 7. This is the exact value. If you want a mixed number or decimal, you can convert it, but the fraction is the precise answer. Final answer: $$\boxed{\frac{759375}{1024}}$$