1. **State the problem:** Simplify the expression $$\frac{(11 \frac{1}{2})^{12} \times 11^{19} \times \sqrt{9}}{27 \times 12^2 \times 11^{11}}$$. 2. **Rewrite mixed number:** Convert $11 \frac{1}{2}$ to an improper fraction or decimal. Here, $11 \frac{1}{2} = 11.5 = \frac{23}{2}$. 3. **Simplify square root:** $\sqrt{9} = 3$. 4. **Substitute and rewrite:** $$\frac{\left(\frac{23}{2}\right)^{12} \times 11^{19} \times 3}{27 \times 12^2 \times 11^{11}}$$ 5. **Simplify powers of 11:** Use the rule $a^m \div a^n = a^{m-n}$: $$11^{19} \div 11^{11} = 11^{19-11} = 11^8$$ 6. **Rewrite denominator powers:** $12^2 = 144$ and $27 = 3^3$. 7. **Rewrite expression:** $$\frac{\left(\frac{23}{2}\right)^{12} \times 11^8 \times 3}{3^3 \times 144}$$ 8. **Simplify the 3's:** $$\frac{3}{3^3} = \frac{3}{27} = \frac{1}{9}$$ So expression becomes: $$\frac{\left(\frac{23}{2}\right)^{12} \times 11^8}{9 \times 144}$$ 9. **Multiply denominator:** $9 \times 144 = 1296$. 10. **Final simplified form:** $$\frac{\left(\frac{23}{2}\right)^{12} \times 11^8}{1296}$$ This is the simplified exact expression. **Optional:** You can leave the answer as is or approximate numerically if needed.