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To solve for \( x \) in the equation:

\[
216^{3x} = 1296^{2x + 4}
\]

we can proceed by expressing both \( 216 \) and \( 1296 \) as powers of a common base.

1. **Express each number as a power of a smaller base:**

   - \( 216 = 6^3 \), so \( 216^{3x} = (6^3)^{3x} = 6^{9x} \).
   - \( 1296 = 6^4 \), so \( 1296^{2x + 4} = (6^4)^{2x + 4} = 6^{8x + 16} \).

   Substituting these into the equation gives:
   \[
   6^{9x} = 6^{8x + 16}
   \]

2. **Set the exponents equal to each other** (since the bases are now the same):
   \[
   9x = 8x + 16
   \]

3. **Solve for \( x \):**
   \[
   9x - 8x = 16
   \]
   \[
   x = 16
   \]

### Final Answer:
\[
x = 16
\]

Would you like further details on this solution or have any additional questions?

Here are some related questions you might find helpful:
1. How would this solution change if we had different exponents for \( x \)?
2. What if the base numbers were prime, such as 3 or 5?
3. How can you use logarithms to solve equations like this when the bases aren't the same?
4. How do you convert exponential equations to logarithmic form?
5. Why is it useful to express numbers as powers of a common base?

**Tip:** Always check if you can rewrite numbers as powers of smaller numbers to simplify equations like this one.

Learn how to solve the exponential equation 216^(3x) = 1296^(2x + 4) by expressing terms with a common base and setting the exponents equal to each other. This solution covers algebraic concepts, properties of exponents, and step-by-step simplification, suitable for high school students in Grades 10-12.

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