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Let's solve the given expression step by step.

The expression provided is:

\[
\frac{\left( \sqrt{2} + \frac{1}{\sqrt{2}} \right) \cdot \left( \sqrt{3} - \frac{1}{\sqrt{3}} \right)}{\sqrt{6}}
\]

### Step 1: Simplify each term inside the parentheses
- The first term inside the parentheses is \( \sqrt{2} + \frac{1}{\sqrt{2}} \). To simplify it:
  
  \[
  \sqrt{2} + \frac{1}{\sqrt{2}} = \sqrt{2} + \frac{\sqrt{2}}{2} = \frac{2\sqrt{2}}{2} + \frac{\sqrt{2}}{2} = \frac{3\sqrt{2}}{2}
  \]

- The second term is \( \sqrt{3} - \frac{1}{\sqrt{3}} \). To simplify it:
  
  \[
  \sqrt{3} - \frac{1}{\sqrt{3}} = \frac{2\sqrt{3}}{3}
  \]

### Step 2: Multiply the simplified terms
Now multiply the simplified terms:

\[
\left( \frac{3\sqrt{2}}{2} \right) \cdot \left( \frac{2\sqrt{3}}{3} \right)
\]

This simplifies to:

\[
\frac{3\sqrt{2} \cdot 2\sqrt{3}}{2 \cdot 3} = \frac{6\sqrt{6}}{6} = \sqrt{6}
\]

### Step 3: Divide by \( \sqrt{6} \)
The entire expression becomes:

\[
\frac{\sqrt{6}}{\sqrt{6}} = 1
\]

### Final Answer:
The result of the operation is **1**. Therefore, the correct answer is **C) 1**.

---

Would you like further details on any specific step or explanation? Here are five related questions you might find useful:

1. How do you rationalize the denominator of a fraction involving square roots?
2. What are some techniques for simplifying radical expressions?
3. Can you explain why \( \sqrt{a} \times \sqrt{b} = \sqrt{ab} \)?
4. How do you approach complex expressions with both addition and multiplication of square roots?
5. What are some strategies to check your work when simplifying complex expressions?

**Tip:** Always simplify inside parentheses first and handle operations step by step to avoid errors!

What is the result of the following expression: \( \frac{\left( \sqrt{2} + \frac{1}{\sqrt{2}} \right) \cdot \left( \sqrt{3} - \frac{1}{\sqrt{3}} \right)}{\sqrt{6}} \)?

Solve a complex expression involving square roots: \( \frac{\left( \sqrt{2} + \frac{1}{\sqrt{2}} \right) \cdot \left( \sqrt{3} - \frac{1}{\sqrt{3}} \right)}{\sqrt{6}} \). Learn the step-by-step process of simplifying radicals and applying the properties of square roots.

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