We need to solve the equation:  

\[
\left( \frac{1}{25} \right)^x = 5^{2x - 1}
\]

### Step 1: Express \( \frac{1}{25} \) as a Power of 5  
Since \( 25 = 5^2 \), we rewrite:

\[
\frac{1}{25} = 5^{-2}
\]

Thus, our equation becomes:

\[
(5^{-2})^x = 5^{2x - 1}
\]

### Step 2: Apply the Power Rule  
Using the exponent rule \( (a^m)^n = a^{m \cdot n} \), we simplify:

\[
5^{-2x} = 5^{2x - 1}
\]

### Step 3: Set Exponents Equal to Each Other  
Since the bases are the same, we equate the exponents:

\[
-2x = 2x - 1
\]

### Step 4: Solve for \( x \)  
1. Add \( 2x \) to both sides:

   \[
   0 = 4x - 1
   \]

2. Add 1 to both sides:

   \[
   1 = 4x
   \]

3. Divide by 4:

   \[
   x = \frac{1}{4}
   \]

### Final Answer:  
\[
\boxed{\frac{1}{4}}
\]

Would you like a detailed breakdown of any step?  

Here are some related questions:  
1. How do exponent rules help in solving exponential equations?  
2. What happens when the bases in an equation are different?  
3. Can this equation be solved using logarithms?  
4. How does changing the base of the exponent affect the solution?  
5. What real-world applications involve exponential equations like this?  

**Tip:** When dealing with exponents, always try to express all terms with the same base to simplify calculations.

Learn how to solve the exponential equation (1/25)^x = 5^(2x - 1) step by step using exponent rules. This problem covers key algebraic concepts such as rewriting bases, applying exponent properties, and solving for x. Suitable for students in Grades 9-11.

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