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The problem asks you to rationalize the denominator of the expression:

\[
\frac{-8}{\sqrt{8x}}
\]

### Steps to solve:
1. **Multiply both the numerator and denominator by \(\sqrt{8x}\)** to eliminate the square root from the denominator:
   \[
   \frac{-8}{\sqrt{8x}} \times \frac{\sqrt{8x}}{\sqrt{8x}} = \frac{-8\sqrt{8x}}{(\sqrt{8x})^2}
   \]

2. Simplify the denominator:
   \[
   (\sqrt{8x})^2 = 8x
   \]
   So the expression becomes:
   \[
   \frac{-8\sqrt{8x}}{8x}
   \]

3. Now simplify the fraction by canceling out the common factor of 8:
   \[
   \frac{-8\sqrt{8x}}{8x} = \frac{-\sqrt{8x}}{x}
   \]

### Final Answer:
\[
\boxed{\frac{-\sqrt{8x}}{x}}
\]

Would you like more details on any part of the process? 

### Related Questions:
1. How do you simplify square roots involving variables?
2. What is the process for rationalizing more complex denominators?
3. How can we factor expressions involving square roots?
4. What happens when variables are under the square root in both the numerator and denominator?
5. What are some common mistakes to avoid when rationalizing the denominator?

### Tip:
When rationalizing the denominator, always check if you can simplify the square root before multiplying.

Rationalize the denominator: -8 / sqrt(8x).

Learn how to rationalize the denominator of -8 / sqrt(8x) by eliminating the square root. This step-by-step solution covers multiplying by the conjugate and simplifying square roots. Perfect for students in Grades 9-12 studying algebra and rationalization.

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