The expression $\frac{2}{1 + \sqrt{3} - \sqrt{2}}$ can be simplified by rationalizing the denominator. Here's how we can do it:

Step 1: Rationalize the denominator

To rationalize the denominator, multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of $1 + \sqrt{3} - \sqrt{2}$ is $1 + \sqrt{3} + \sqrt{2}$.

So, multiply the expression by $\frac{1 + \sqrt{3} + \sqrt{2}}{1 + \sqrt{3} + \sqrt{2}}$:

$\frac{2}{1 + \sqrt{3} - \sqrt{2}} \times \frac{1 + \sqrt{3} + \sqrt{2}}{1 + \sqrt{3} + \sqrt{2}} = \frac{2(1 + \sqrt{3} + \sqrt{2})}{(1 + \sqrt{3})^2 - (\sqrt{2})^2}$

Step 2: Simplify the denominator

The denominator now becomes:

$(1 + \sqrt{3})^2 - (\sqrt{2})^2$

First, expand the square:

$(1 + \sqrt{3})^2 = 1 + 2\sqrt{3} + 3 = 4 + 2\sqrt{3}$

Subtract the square of $\sqrt{2}$:

$4 + 2\sqrt{3} - 2 = 2 + 2\sqrt{3}$

So the denominator simplifies to:

$2(1 + \sqrt{3})$

Step 3: Simplify the entire expression

Now the expression becomes:

$\frac{2(1 + \sqrt{3} + \sqrt{2})}{2(1 + \sqrt{3})}$

You can cancel out the common factor of 2 from the numerator and denominator:

$\frac{1 + \sqrt{3} + \sqrt{2}}{1 + \sqrt{3}}$

Final Result:

The simplified expression is:

$\frac{1 + \sqrt{3} + \sqrt{2}}{1 + \sqrt{3}}$

Would you like a more detailed explanation, or do you have any questions?

Related Questions:

1. How do you rationalize more complex expressions involving square roots?
2. What is the general formula for the difference of squares?
3. How does multiplying by the conjugate help in simplifying expressions?
4. Can you always cancel out terms in the numerator and denominator directly?
5. What happens if the denominator has a sum of squares instead?

Tip: Rationalizing the denominator can make expressions much easier to work with, especially when dealing with square roots.