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

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To simplify the expression:

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

**Step 1: Apply the exponent to the numerator.**

$$
\left(3 x^{2} y^{2}\right)^{2} = 3^{2} \cdot x^{2 \times 2} \cdot y^{2 \times 2} = 9 x^{4} y^{4}
$$

**Step 2: Rewrite the entire expression with the simplified numerator.**

$$
\frac{9 x^{4} y^{4}}{27 x^{-2} y^{3}}
$$

**Step 3: Simplify the coefficients and variables separately.**

- **Coefficients:**
  $$
  \frac{9}{27} = \frac{1}{3}
  $$
  
- **For $$x$$:**
  $$
  x^{4} \div x^{-2} = x^{4 - (-2)} = x^{6}
  $$
  
- **For $$y$$:**
  $$
  y^{4} \div y^{3} = y^{4 - 3} = y^{1} = y
  $$

**Step 4: Combine the simplified parts.**

$$
\frac{1}{3} x^{6} y
$$

So, the simplified form of the expression is:

$$
\frac{x^{6} y}{3}
$$

**Final Answer:**

$$\dfrac{x^{6}\, y}{3}$$
$$\dfrac{x^{6}\, y}{3}$$

Simplify:

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