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The given equation is \( x^x = 256 \), and we need to find the value of \( x \).

### Step-by-step Solution:
We know that \( 256 = 2^8 \). So, let's test values of \( x \) to see if we can find a match.

1. **Testing \( x = 4 \)**:
   \[
   x^x = 4^4 = 256
   \]
   This is correct, so the value of \( x \) is \( \boxed{4} \).

Would you like a more detailed explanation or further steps?

### Related Questions:
1. What would happen if \( x^x = 1024 \)?
2. Can \( x^x \) have more than one solution for some values?
3. How does \( x^x \) grow as \( x \) increases?
4. What is the solution for \( x^x = 512 \)?
5. What if the equation were \( x^{x+1} = 256 \)?

### Tip:
When you encounter equations like \( x^x = k \), try expressing \( k \) as powers of simpler numbers to find solutions more easily.

Solve the exponential equation x^x = 256 with a step-by-step solution. The problem involves testing values of x and using properties of powers to find x = 4. This topic is suitable for students in Grades 9-11, covering key concepts in exponential equations.

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