1. **State the problem:** Simplify the expression $$\frac{(11 \times 2 \times 3) \times 12 \times 11 \times 9 \times \sqrt{9}}{2 \times 7 \times (12 \times 1) \times 2 \times 11 \times 1}$$. 2. **Recall important rules:** - Multiplication is associative and commutative, so we can rearrange terms. - $$\sqrt{9} = 3$$ because 3 squared is 9. - Simplify numerator and denominator separately, then reduce the fraction by canceling common factors. 3. **Simplify the numerator:** $$11 \times 2 \times 3 = 66$$ So numerator becomes: $$66 \times 12 \times 11 \times 9 \times 3$$ Calculate stepwise: $$66 \times 12 = 792$$ $$792 \times 11 = 8712$$ $$8712 \times 9 = 78408$$ $$78408 \times 3 = 235224$$ 4. **Simplify the denominator:** $$12 \times 1 = 12$$ So denominator becomes: $$2 \times 7 \times 12 \times 2 \times 11 \times 1$$ Calculate stepwise: $$2 \times 7 = 14$$ $$14 \times 12 = 168$$ $$168 \times 2 = 336$$ $$336 \times 11 = 3696$$ 5. **Form the fraction:** $$\frac{235224}{3696}$$ 6. **Simplify the fraction by dividing numerator and denominator by their greatest common divisor (GCD):** Calculate GCD of 235224 and 3696. 7. **Divide numerator and denominator by 12 first:** $$\frac{\cancel{235224}^{12 \times 19602}}{\cancel{3696}^{12 \times 308}} = \frac{19602}{308}$$ 8. **Divide numerator and denominator by 7:** $$\frac{\cancel{19602}^{7 \times 2800 + 2}}{\cancel{308}^{7 \times 44}} = \frac{2800 + \frac{2}{7}}{44}$$ Since 2/7 is not an integer, divide by 2 instead. 9. **Divide numerator and denominator by 2:** $$\frac{\cancel{19602}^{2 \times 9801}}{\cancel{308}^{2 \times 154}} = \frac{9801}{154}$$ 10. **Check if 9801 and 154 share any common factors:** - 9801 = 99^2 (since 99 x 99 = 9801) - 154 = 2 x 7 x 11 No common factors, so fraction is simplified. **Final answer:** $$\boxed{\frac{9801}{154}}$$