Simplify the expression. \[ \left(\frac{3 x^{3} y^{5}}{2 x^{2} y}\right)^{4}=\square \]

Let's simplify the expression step-by-step: First, simplify the inside of the parentheses: \[ \frac{3 x^{3} y^{5}}{2 x^{2} y} = \frac{3}{2} \cdot \frac{x^{3}}{x^{2}} \cdot \frac{y^{5}}{y} \] For the \(x\) terms, we subtract exponents: \[ \frac{x^{3}}{x^{2}} = x^{3-2} = x^{1} = x \] For the \(y\) terms, we also subtract exponents: \[ \frac{y^{5}}{y} = \frac{y^{5}}{y^{1}} = y^{5-1} = y^{4} \] Now, we can rewrite the expression: \[ \frac{3 x^{3} y^{5}}{2 x^{2} y} = \frac{3}{2} xy^{4} \] Now, we take this simplified expression and raise it to the power of 4: \[ \left(\frac{3}{2} xy^{4}\right)^{4} = \left(\frac{3}{2}\right)^{4} (x)^{4} (y^{4})^{4} \] Calculating each part: \[ \left(\frac{3}{2}\right)^{4} = \frac{3^{4}}{2^{4}} = \frac{81}{16} \] \[ x^{4} = x^{4} \] \[ (y^{4})^{4} = y^{16} \] Putting it all together: \[ \left(\frac{3 x^{3} y^{5}}{2 x^{2} y}\right)^{4} = \frac{81}{16} x^{4} y^{16} \] So, the simplified expression is: \[ \frac{81}{16} x^{4} y^{16} \] Therefore, the final answer is: \[ \frac{81}{16} x^{4} y^{16} = \square \]