To solve the equation \( 3^x + 4^x = 25 \), we can begin by testing small integer values of \( x \) to check if any satisfy the equation.

### Step-by-step approach:
1. **Substitute \( x = 2 \):**
   \[
   3^2 + 4^2 = 9 + 16 = 25
   \]
   So, \( x = 2 \) is a solution.

2. **Check for other solutions:**
   Since both \( 3^x \) and \( 4^x \) are increasing functions, the sum \( 3^x + 4^x \) will increase as \( x \) increases. Similarly, for negative \( x \), the values will decrease, and it will be hard for the sum to reach 25 again. Thus, \( x = 2 \) is likely the only real solution.

### Final answer:
The solution to the equation is \( x = 2 \).

Would you like more details or have any questions?

Here are 5 related questions to explore further:
1. What happens if we change the base of the exponents in a similar equation?
2. Can we solve \( 2^x + 5^x = 30 \) in the same way?
3. What would happen if the right-hand side of the equation was greater or smaller than 25?
4. How do you graphically represent equations like \( 3^x + 4^x = 25 \)?
5. How can we solve \( 3^x + 4^x = 100 \)?

**Tip:** When dealing with exponential equations, testing small integer values often helps identify solutions quickly!

Discover how to solve the exponential equation 3^x + 4^x = 25 by testing small integer values and analyzing function behavior. This problem demonstrates key concepts in algebra and exponential functions, making it suitable for students in Grades 9-12.

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