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The given expression simplifies to:

\[
\log_5 \left( \frac{5^{-2} \cdot \sqrt{625}}{25 \cdot \sqrt{125}} \right) = \frac{-7}{2}
\]

Let me know if you need further clarification or a detailed breakdown of each step.

Here are some related questions for further exploration:

1. How do we convert logarithmic expressions to exponential form?
2. What are the properties of logarithms that help simplify complex expressions?
3. How can we simplify roots and powers involving large numbers?
4. What happens when the base of a logarithm is changed?
5. How do negative exponents affect the value of an expression?

**Tip**: Always simplify square roots and powers separately before combining terms for easier calculations.

Solve the logarithmic expression log_5((5^(-2) * sqrt(625)) / (25 * sqrt(125)))

Learn how to solve and simplify a logarithmic expression involving exponents and square roots. This problem covers logarithm properties, exponent rules, and radical simplifications. Suitable for students in Grades 10-12.

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