1. The problem is to simplify the expression $$\frac{1}{\sqrt{3} - 1}$$. 2. To simplify, multiply numerator and denominator by the conjugate of the denominator $$\sqrt{3} + 1$$ to rationalize it: $$\frac{1}{\sqrt{3} - 1} \times \frac{\sqrt{3} + 1}{\sqrt{3} + 1} = \frac{\sqrt{3} + 1}{(\sqrt{3} - 1)(\sqrt{3} + 1)}$$ 3. Use the difference of squares formula for the denominator: $$(\sqrt{3})^2 - 1^2 = 3 - 1 = 2$$ 4. So the expression becomes: $$\frac{\sqrt{3} + 1}{2}$$ 5. This is the simplified form of the original expression. Final answer: $$\frac{\sqrt{3} + 1}{2}$$