We are tasked with simplifying the expression:

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

Step 1: Combine like terms under a common denominator

First, treat the square roots and constants together by rewriting the denominator:

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

Now, we aim to rationalize the denominator, which means eliminating the square roots from it.

Step 2: Rationalize the denominator

To rationalize the denominator, multiply both the numerator and denominator by the conjugate of the denominator. The conjugate is obtained by reversing the sign between the terms involving square roots. In this case, the conjugate is:

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

Now multiply the original expression by this conjugate over itself:

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

Step 3: Apply the difference of squares

Now simplify the denominator using the difference of squares formula $(a - b)(a + b) = a^2 - b^2$:

[ \left[ (\sqrt{3} - 1) - \sqrt{2} \right] \left[ (\sqrt{3} - 1) + \sqrt{2} \right] = (\sqrt{3} - 1)^2 - (\sqrt{2})^2 ]

First, expand $(\sqrt{3} - 1)^2$:

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

Now, subtract $(\sqrt{2})^2 = 2$:

$(4 - 2\sqrt{3}) - 2 = 2 - 2\sqrt{3}$

So, the denominator becomes:

$2 - 2\sqrt{3}$

Step 4: Simplify the expression

Now we have:

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

To simplify this further, factor out a 2 from the denominator:

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

Final Answer:

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

Let me know if you'd like me to clarify any step.

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Questions to Explore:

1. What is the difference between rationalizing a numerator and a denominator?
2. How does the difference of squares formula work for radical terms?
3. Can we further simplify the expression without expanding terms?
4. How do square roots affect simplification in algebraic expressions?
5. What are other methods of simplifying complex fractions with radicals?

Tip: Always check if factoring common terms from numerators or denominators can make simplifications easier!