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To simplify the expression \(\frac{\sqrt{-102}}{\sqrt{6}}\), let's break down the steps:

1. **Identify the Complex Component**: Since the square root of \(-102\) involves a negative under the square root, we rewrite it as:
   \[
   \sqrt{-102} = \sqrt{102} \cdot i
   \]
   where \( i \) is the imaginary unit, defined as \( i = \sqrt{-1} \).

2. **Rewrite the Expression**: Substitute this into the original expression:
   \[
   \frac{\sqrt{-102}}{\sqrt{6}} = \frac{\sqrt{102} \cdot i}{\sqrt{6}}
   \]

3. **Combine the Radicals**: We can combine the square roots in the numerator and denominator:
   \[
   = \sqrt{\frac{102}{6}} \cdot i
   \]

4. **Simplify the Fraction Under the Radical**: Divide 102 by 6:
   \[
   \frac{102}{6} = 17
   \]
   so the expression simplifies to:
   \[
   \sqrt{17} \cdot i
   \]

5. **Final Answer**: Thus, the simplified form of the expression is:
   \[
   \sqrt{17} \cdot i \quad \text{or simply} \quad i\sqrt{17}
   \]

Would you like further details on these steps?

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Here are some related questions you might find useful:

1. How do you simplify expressions with imaginary numbers?
2. What are the rules for dividing square roots?
3. How is the imaginary unit \(i\) defined, and why does it exist in mathematics?
4. What is the process for simplifying fractions under a square root?
5. How can square roots of negative numbers be expressed using complex numbers?

**Tip**: When simplifying square roots of negative numbers, always remember to factor out \(i\) (as in \(\sqrt{-1}\)) to handle the imaginary component separately.

Simplify the expression \( \frac{\sqrt{-102}}{\sqrt{6}} \).

Learn how to simplify the expression \( \frac{\sqrt{-102}}{\sqrt{6}} \) using complex numbers and radicals. This solution includes step-by-step instructions and covers important algebraic concepts for high school students.

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