1. **Problem:** Evaluate $f(1,1)$ for $f(x,y) = x^2 e^{3xy}$. 2. **Formula:** The function is given as $f(x,y) = x^2 e^{3xy}$. 3. **Step 1:** Substitute $x=1$ and $y=1$ into the function: $$f(1,1) = 1^2 e^{3 \cdot 1 \cdot 1} = e^3$$ 4. **Step 2:** Calculate the value: Since $e^3$ is approximately $20.0855$, the exact value is $e^3$. **Final answer:** $f(1,1) = e^3$