Rationalise the den a) \( \frac{1+\sqrt{2}}{1-\sqrt{2}} \)

To rationalize the denominator of the expression: \[ \frac{1 + \sqrt{2}}{1 - \sqrt{2}} \] follow these steps: ### Step 1: Multiply by the Conjugate The conjugate of the denominator \(1 - \sqrt{2}\) is \(1 + \sqrt{2}\). Multiply both the numerator and the denominator by this conjugate to eliminate the radical in the denominator: \[ \frac{1 + \sqrt{2}}{1 - \sqrt{2}} \times \frac{1 + \sqrt{2}}{1 + \sqrt{2}} = \frac{(1 + \sqrt{2}) \times (1 + \sqrt{2})}{(1 - \sqrt{2}) \times (1 + \sqrt{2})} \] ### Step 2: Expand the Numerator and Denominator **Numerator:** \[ (1 + \sqrt{2})^2 = 1^2 + 2 \times 1 \times \sqrt{2} + (\sqrt{2})^2 = 1 + 2\sqrt{2} + 2 = 3 + 2\sqrt{2} \] **Denominator:** \[ (1 - \sqrt{2})(1 + \sqrt{2}) = 1^2 - (\sqrt{2})^2 = 1 - 2 = -1 \] ### Step 3: Simplify the Expression Now, the expression becomes: \[ \frac{3 + 2\sqrt{2}}{-1} = - (3 + 2\sqrt{2}) = -3 - 2\sqrt{2} \] ### Final Answer After rationalizing, the expression simplifies to: \[ -3 - 2\sqrt{2} \]