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The given problem states that:

\[
8^x = 5
\]

You are asked to find the value of \( 8^{-2x} \).

### Solution Steps:

1. We know that \( 8^x = 5 \). This means that the expression for \( 8^{-2x} \) can be written using this information.

2. Recall that \( 8^{-2x} \) can be rewritten as:
   \[
   8^{-2x} = (8^x)^{-2}
   \]

3. Since \( 8^x = 5 \), we substitute this into the expression:
   \[
   (8^x)^{-2} = 5^{-2}
   \]

4. Simplifying \( 5^{-2} \):
   \[
   5^{-2} = \frac{1}{5^2} = \frac{1}{25}
   \]

Thus, \( 8^{-2x} = \frac{1}{25} \).

The final answer is \( \boxed{\frac{1}{25}} \).

---

Would you like any further details or explanations? Here are some follow-up questions to consider:

1. How can you express negative exponents in terms of reciprocals?
2. Can you explain why \( 8^{-2x} \) is equivalent to \( (8^x)^{-2} \)?
3. How do you simplify powers of fractions like \( 5^{-2} \)?
4. What are other ways to manipulate exponential equations?
5. How does this problem change if \( 8^x = 6 \) instead of 5?

**Tip:** When working with exponents, remember that negative exponents represent reciprocals, and exponent rules can simplify complex expressions.

Learn how to solve an exponential equation where 8^x = 5, and find the value of 8^-2x. This problem covers key concepts in exponents and algebra, suitable for students in Grades 8-10.

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