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.

Learn how to simplify the exponential expression e^3ln(x^2) using logarithmic properties and exponential-logarithmic identities. This detailed solution walks through key concepts in algebra and is suitable for students in Grades 9-11.

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