The expression that needs to be simplified is:

$\frac{3\sqrt{2}}{\sqrt{6} - \sqrt{3}} - \frac{4\sqrt{3}}{\sqrt{6} - \sqrt{2}} + \frac{2\sqrt{3}}{\sqrt{6} + 2}$

Let's break this down step by step.

Step 1: Rationalize the Denominators

The first two fractions involve square roots in the denominator, so we will rationalize each of them.

• For the first term, $\frac{3\sqrt{2}}{\sqrt{6} - \sqrt{3}}$, multiply the numerator and denominator by $\sqrt{6} + \sqrt{3}$.

• For the second term, $\frac{4\sqrt{3}}{\sqrt{6} - \sqrt{2}}$, multiply the numerator and denominator by $\sqrt{6} + \sqrt{2}$.

The third term already has a rational denominator, so we leave that as is.

Let me compute these steps for you.After rationalizing the denominators, the simplified terms are:

1. $\frac{3\sqrt{2}}{\sqrt{6} - \sqrt{3}} = \sqrt{6} + 2\sqrt{3}$
2. $\frac{4\sqrt{3}}{\sqrt{6} - \sqrt{2}} = \sqrt{6} + 3\sqrt{2}$
3. The third term remains: $\frac{2\sqrt{3}}{\sqrt{6} + 2} = \frac{2\sqrt{3}}{2 + \sqrt{6}}$

Now, we combine these three simplified terms:

$(\sqrt{6} + 2\sqrt{3}) - (\sqrt{6} + 3\sqrt{2}) + \frac{2\sqrt{3}}{2 + \sqrt{6}}$

I will proceed to simplify the expression further.The entire expression simplifies to 0.

This means the given expression:

$\frac{3\sqrt{2}}{\sqrt{6} - \sqrt{3}} - \frac{4\sqrt{3}}{\sqrt{6} - \sqrt{2}} + \frac{2\sqrt{3}}{\sqrt{6} + 2}$

is equivalent to zero after simplification.

Would you like further details or have any other questions? Here are a few related questions for practice:

1. How do you rationalize the denominator of a fraction?
2. What is the process to simplify square root expressions?
3. Why is multiplying by the conjugate useful in rationalization?
4. Can you simplify $\frac{5}{\sqrt{2} + 1}$ using a similar method?
5. How do different square root expressions behave when combined?

Tip: Rationalizing denominators often involves multiplying by the conjugate to eliminate square roots from the denominator.