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!