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

To simplify the expression \( \frac{\left(3 x^{2} y^{2}\right)^{2}}{3^{3} x^{2} y^{3}} \), let's break it down step by step. First, simplify the numerator: \[ \left(3 x^{2} y^{2}\right)^{2} = 3^{2} \cdot (x^{2})^{2} \cdot (y^{2})^{2} = 9 x^{4} y^{4} \] Now, we can rewrite the original expression as: \[ \frac{9 x^{4} y^{4}}{3^{3} x^{2} y^{3}} \] Next, simplify the denominator: \[ 3^{3} = 27 \] So, we have: \[ \frac{9 x^{4} y^{4}}{27 x^{2} y^{3}} \] Now, let's simplify the fraction by dividing both the numerator and the denominator by 9: \[ \frac{9 x^{4} y^{4}}{27 x^{2} y^{3}} = \frac{x^{4} y^{4}}{3 x^{2} y^{3}} \] Now, simplify the \(x\) and \(y\) terms in the fraction: 1. For \(x\): \(x^{4} / x^{2} = x^{2}\) 2. For \(y\): \(y^{4} / y^{3} = y^{1} = y\) Putting it all together, we have: \[ \frac{x^{2} y}{3} \] Thus, the simplified expression is: \[ \frac{x^{2} y}{3} \]