To solve the equation \(6^x = 36^{2-x}\), follow these steps:

1. **Rewrite \(36\) in terms of \(6\):**
   \[
   36 = 6^2
   \]
   Thus, the equation becomes:
   \[
   6^x = (6^2)^{2-x}
   \]

2. **Simplify the right side using the power of a power property:**
   \[
   (6^2)^{2-x} = 6^{2(2-x)} = 6^{4-2x}
   \]
   Now the equation is:
   \[
   6^x = 6^{4-2x}
   \]

3. **Since the bases are the same, set the exponents equal to each other:**
   \[
   x = 4 - 2x
   \]

4. **Solve for \(x\):**
   \[
   x + 2x = 4
   \]
   \[
   3x = 4
   \]
   \[
   x = \frac{4}{3}
   \]

So, the solution to the equation \(6^x = 36^{2-x}\) is \(x = \frac{4}{3}\).

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

Here are 8 related questions to expand on this information:
1. How do you solve equations with different bases?
2. What are the properties of exponents used in this solution?
3. How can you verify the solution to an exponential equation?
4. What is the power of a power property?
5. How do you simplify expressions involving exponents?
6. Can you provide another example of solving an exponential equation?
7. How would the solution change if the equation were \(6^x = 36^{x-2}\)?
8. What are logarithms and how can they be used to solve exponential equations?

**Tip:** When dealing with exponential equations, try to express all terms with the same base to simplify solving the equation.

Learn how to solve the exponential equation 6^x = 36^(2-x) with a detailed step-by-step solution. This problem covers key concepts in exponential equations and the power of a power property. Suitable for students in Grades 10-12.

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