You're on a roll — Step 1: Simplify the numerator. The numerator is $\sqrt{9x^4y^2}$. We can simplify each term under the square root: $$\sqrt{9x^4y^2} = \sqrt{9} \cdot \sqrt{x^4} \cdot \sqrt{y^2}$$ $$= 3 \cdot x^{4/2} \cdot y^{2/2}$$ $$= 3x^2y$$ Step 2: Simplify the denominator. The denominator is $\sqrt[3]{27x^6y^{12}}$. We can simplify each term under the cube root: $$\sqrt[3]{27x^6y^{12}} = \sqrt[3]{27} \cdot \sqrt[3]{x^6} \cdot \sqrt[3]{y^{12}}$$ $$= 3 \cdot x^{6/3} \cdot y^{12/3}$$ $$= 3x^2y^4$$ Step 3: Divide the simplified numerator by the simplified denominator. Now, substitute the simplified expressions back into the fraction: $$\frac{3x^2y}{3x^2y^4}$$ Cancel out the common terms $3$ and $x^2$: $$= \frac{y}{y^4}$$ Using the rule of exponents $\frac{a^m}{a^n} = a^{m-n}$: $$= y^{1-4}$$ $$= y^{-3}$$ This can also be written as: $$= \frac{1}{y^3}$$ The simplified expression is $\frac{1}{y^3}$. $$\boxed{\frac{1}{y^3}}$$