Simplify: \( \frac{\left(3 x^{2} y^{2}\right)^{2}}{3^{3} x^{-2} y^{3}} \)

To simplify the expression \( \frac{\left(3 x^{2} y^{2}\right)^{2}}{3^{3} x^{-2} y^{3}} \), we start by simplifying the numerator and the denominator. First, we simplify the numerator: \[ \left(3 x^{2} y^{2}\right)^{2} = 3^{2} (x^{2})^{2} (y^{2})^{2} = 9 x^{4} y^{4} \] Next, we simplify the denominator: \[ 3^{3} x^{-2} y^{3} = 27 x^{-2} y^{3} \] Now we have: \[ \frac{9 x^{4} y^{4}}{27 x^{-2} y^{3}} \] We can simplify the fraction by dividing the coefficients and the variables separately. For the coefficients: \[ \frac{9}{27} = \frac{1}{3} \] For \( x \): \[ \frac{x^{4}}{x^{-2}} = x^{4 - (-2)} = x^{4 + 2} = x^{6} \] For \( y \): \[ \frac{y^{4}}{y^{3}} = y^{4 - 3} = y^{1} = y \] Now, putting it all together, we have: \[ \frac{1}{3} x^{6} y \] Thus, the simplified form of the original expression is: \[ \frac{x^{6} y}{3} \]