QUESTION 2 2.1 Without using a calculator, rationalise the denominator and simplify: \( \frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}+\sqrt{2}} \)

To rationalise the denominator of \( \frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}+\sqrt{2}} \), multiply the numerator and denominator by the conjugate of the denominator, \( \sqrt{3}-\sqrt{2} \): \[ \frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}+\sqrt{2}} \cdot \frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}-\sqrt{2}} = \frac{(\sqrt{3}-\sqrt{2})^2}{(\sqrt{3})^2 - (\sqrt{2})^2} \] Calculating the denominator: \[ (\sqrt{3})^2 - (\sqrt{2})^2 = 3 - 2 = 1 \] Calculating the numerator: \[ (\sqrt{3} - \sqrt{2})^2 = 3 - 2\sqrt{6} + 2 = 5 - 2\sqrt{6} \] Thus, the expression simplifies to: \[ 5 - 2\sqrt{6} \] So, the final answer is: \[ 5 - 2\sqrt{6} \]