Simplify the exponential expression. \[ \left(3 y^{4}\right)^{2} \]

To simplify the expression \((3 y^{4})^{2}\), you need to apply the power of a product rule and the power of a power rule. First, raise the coefficient and the variable to the power of 2: \[ (3 y^{4})^{2} = 3^{2} (y^{4})^{2} \] Calculating each part gives: \[ 3^{2} = 9 \quad \text{and} \quad (y^{4})^{2} = y^{4 \cdot 2} = y^{8} \] Combining these results, we get: \[ \left(3 y^{4}\right)^{2} = 9 y^{8} \] Thus, the simplified expression is: \[ \boxed{9 y^{8}} \]