The expression $e^{3 \ln(x^2)}$ can be simplified using logarithmic properties. Here's the step-by-step simplification:

1. First, apply the logarithmic property $\ln(a^b) = b \ln(a)$: $e^{3 \ln(x^2)} = e^{3 \cdot 2 \ln(x)} = e^{6 \ln(x)}$

2. Now, apply the exponential-logarithmic identity $e^{\ln(a)} = a$: $e^{6 \ln(x)} = x^6$

Thus, the simplified form of $e^{3 \ln(x^2)}$ is:

$x^6$

Would you like more details, or do you have any further questions?

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Here are five related questions for practice:

1. Simplify $e^{2 \ln(x^3)}$.
2. Simplify $e^{\ln(x^5)}$.
3. Simplify $\ln(e^{3x})$.
4. What is $e^{\ln(2x^2)}$?
5. Solve $e^{x \ln(4)} = 16$.

Tip: Remember that $e^{\ln(a)} = a$ is a useful identity for simplifying expressions involving exponentials and logarithms.