To simplify the expression \(\frac{5^x (5^h - 1)}{h}\), we will first look at the structure and see if there are any common terms or properties of exponents that can be applied.

### Step-by-Step Simplification:

1. **Expression:**
   \[
   \frac{5^x (5^h - 1)}{h}
   \]

2. **Distribute \(5^x\):**
   The expression inside the parentheses can be multiplied by \(5^x\):
   \[
   = \frac{5^x \cdot 5^h - 5^x}{h}
   \]

   Since \(5^x \cdot 5^h = 5^{x + h}\), the expression becomes:
   \[
   = \frac{5^{x + h} - 5^x}{h}
   \]

3. **Expression After Distribution:**
   This expression is now simplified to a form that might be recognized as a difference quotient, particularly:
   \[
   \frac{5^{x + h} - 5^x}{h}
   \]

### Interpretation:

- The expression \(\frac{5^{x + h} - 5^x}{h}\) is the definition of the derivative of the function \(f(x) = 5^x\) using the limit definition:
  \[
  \lim_{h \to 0} \frac{5^{x + h} - 5^x}{h} = f'(x)
  \]

- If we are not taking the limit as \(h \to 0\), the expression can still be used to represent the average rate of change of \(5^x\) over an interval.

### Conclusion:

The expression simplifies to:
\[
\frac{5^{x + h} - 5^x}{h}
\]
without further reduction unless additional context, such as a limit process, is applied.

Would you like further details on how this relates to derivatives, or have any questions?  

### Related Questions:
1. What is the derivative of \(5^x\) using the definition of the derivative?
2. How can you apply properties of exponents in expressions involving differences?
3. What are difference quotients and why are they important in calculus?
4. How do exponential functions behave as their exponents change?
5. What is the limit of \(\frac{5^{x + h} - 5^x}{h}\) as \(h \to 0\)?

**Tip:** Remember that expressions involving exponents often simplify with the application of properties like \(a^m \cdot a^n = a^{m+n}\).

Explore the detailed step-by-step solution to simplify the expression 5^x * (5^h - 1) / h. Learn about properties of exponents, the definition of derivatives, and the application of difference quotients in calculus. Suitable for Grades 11-12.

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