2. a) Express \( \frac{\sqrt{3}+1}{\sqrt{3}-1}+\sqrt{3}-1 \) in the form \( a+b \sqrt{3} \) where \( a \) and \( b \) are rational numbers. b) Express \( \frac{5}{3-2 \sqrt{3}} \) in the form \( a+b \sqrt{3} \) where \( a \) and \( b \) are rational numbers.

Let's solve each part step by step. ### Problem 2.a) **Express \( \frac{\sqrt{3}+1}{\sqrt{3}-1} + \sqrt{3} - 1 \) in the form \( a + b\sqrt{3} \):** 1. **Rationalize the first term:** \[ \frac{\sqrt{3} + 1}{\sqrt{3} - 1} \times \frac{\sqrt{3} + 1}{\sqrt{3} + 1} = \frac{(\sqrt{3} + 1)^2}{(\sqrt{3})^2 - 1^2} = \frac{3 + 2\sqrt{3} + 1}{3 - 1} = \frac{4 + 2\sqrt{3}}{2} = 2 + \sqrt{3} \] 2. **Add the remaining terms:** \[ (2 + \sqrt{3}) + (\sqrt{3} - 1) = 2 - 1 + 2\sqrt{3} = 1 + 2\sqrt{3} \] **Answer for part a:** \[ 1 + 2\sqrt{3} \] ### Problem 2.b) **Express \( \frac{5}{3 - 2\sqrt{3}} \) in the form \( a + b\sqrt{3} \):** 1. **Rationalize the denominator:** \[ \frac{5}{3 - 2\sqrt{3}} \times \frac{3 + 2\sqrt{3}}{3 + 2\sqrt{3}} = \frac{5(3 + 2\sqrt{3})}{9 - 12} = \frac{15 + 10\sqrt{3}}{-3} = -5 - \frac{10}{3}\sqrt{3} \] **Answer for part b:** \[ -5 - \frac{10}{3}\sqrt{3} \] **Final Answers:** a) \(1 + 2\,\sqrt{3}\) b) \(-5 - \dfrac{10}{3}\, \sqrt{3}\)