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 separately. 1. **Numerator**: \[ \left(3 x^{2} y^{2}\right)^{2} = 3^{2} (x^{2})^{2} (y^{2})^{2} = 9 x^{4} y^{4} \] 2. **Denominator**: \[ 3^{3} x^{-2} y^{3} = 27 x^{-2} y^{3} \] Now, substituting the simplified numerator and denominator into the original expression gives us: \[ \frac{9 x^{4} y^{4}}{27 x^{-2} y^{3}} \] Next, we can simplify the fraction: \[ \frac{9}{27} = \frac{1}{3} \] For the \(x\) terms: \[ \frac{x^{4}}{x^{-2}} = x^{4 - (-2)} = x^{4 + 2} = x^{6} \] For the \(y\) terms: \[ \frac{y^{4}}{y^{3}} = y^{4 - 3} = y^{1} = y \] Putting it all together, we get: \[ \frac{1}{3} x^{6} y \] Thus, the simplified expression is: \[ \frac{x^{6} y}{3} \]