https://file.aiquickdraw.com/mathbot/file/1726018768385rcj8575t.jpg

The equation provided is:

\[
16^{2x-2} = 64^6
\]

### Step-by-Step Solution:

1. **Express both sides with the same base**:
   - We know that \(16 = 2^4\) and \(64 = 2^6\). So, rewrite the equation:
   
   \[
   (2^4)^{2x-2} = (2^6)^6
   \]
   
2. **Simplify the exponents**:
   Using the property \((a^m)^n = a^{m \cdot n}\), we can simplify both sides:
   
   \[
   2^{4(2x-2)} = 2^{6 \cdot 6}
   \]
   This simplifies to:
   
   \[
   2^{8x - 8} = 2^{36}
   \]
   
3. **Set the exponents equal**:
   Since the bases are the same, the exponents must be equal. Thus:
   
   \[
   8x - 8 = 36
   \]
   
4. **Solve for \(x\)**:
   - Add 8 to both sides:
   
     \[
     8x = 44
     \]
   
   - Divide by 8:
   
     \[
     x = \frac{44}{8} = 5.5
     \]

Thus, the solution is \(x = 5.5\).

---

Would you like further details on any step or have any other questions? 

Here are 5 related questions to explore:
1. How do you solve exponential equations with different bases?
2. What are the laws of exponents, and how do they apply to this problem?
3. Can we solve this problem using logarithms instead?
4. How can you simplify powers with fractional exponents?
5. What happens if the bases are not powers of the same number?

**Tip:** Always try to express both sides of an exponential equation using the same base to simplify solving.

Learn how to solve the exponential equation 16^(2x-2) = 64^6 by expressing both sides with the same base and using the laws of exponents. Follow the step-by-step solution to find x = 5.5. This problem is suitable for students in Grades 9-11.

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