May 2026

Find the Value Find the value \[ x = 3 + 2\sqrt{2} \] Solution: \[ x = (\sqrt{2} + 1)^2 \Rightarrow \sqrt{x} = \sqrt{2} + 1 \] \[ \frac{1}{\sqrt{x}} = \frac{1}{\sqrt{2} + 1} \times \frac{\sqrt{2} – 1}{\sqrt{2} – 1} = \sqrt{2} – 1 \] \[ \sqrt{x} + \frac{1}{\sqrt{x}} = (\sqrt{2} + 1) + (\sqrt{2} – […]

Simplify Surd Simplify \[ \sqrt{3 – 2\sqrt{2}} \] Solution: \[ \sqrt{3 – 2\sqrt{2}} = \sqrt{a} – \sqrt{b} \] \[ (\sqrt{a} – \sqrt{b})^2 = a + b – 2\sqrt{ab} \] \[ a + b = 3, \quad 2\sqrt{ab} = 2\sqrt{2} \Rightarrow ab = 2 \] \[ a = 2, \quad b = 1 \] \[ \therefore

Simplify Surd Simplify \[ \sqrt{3 + 2\sqrt{2}} \] Solution: \[ \sqrt{3 + 2\sqrt{2}} = \sqrt{a} + \sqrt{b} \] \[ (\sqrt{a} + \sqrt{b})^2 = a + b + 2\sqrt{ab} \] \[ a + b = 3, \quad 2\sqrt{ab} = 2\sqrt{2} \Rightarrow ab = 2 \] \[ a = 1, \quad b = 2 \] \[ \therefore

Find the Rationalisation Factor Find the rationalisation factor \[ \sqrt{5} – 2 \] Solution: \[ \text{Rationalisation factor of } (a – b) = (a + b) \] \[ \therefore \text{Rationalisation factor of } (\sqrt{5} – 2) = \sqrt{5} + 2 \] Next Question / Full Exercise

Find the Value Find the value \[ x = 2 + \sqrt{3} \] Solution: \[ \frac{1}{x} = \frac{1}{2 + \sqrt{3}} \times \frac{2 – \sqrt{3}}{2 – \sqrt{3}} \] \[ = \frac{2 – \sqrt{3}}{4 – 3} = 2 – \sqrt{3} \] \[ x + \frac{1}{x} = (2 + \sqrt{3}) + (2 – \sqrt{3}) \] \[ = 4

Find the Value Find the value \[ a = \sqrt{2} + 1 \] Solution: \[ \frac{1}{a} = \frac{1}{\sqrt{2} + 1} \times \frac{\sqrt{2} – 1}{\sqrt{2} – 1} \] \[ = \frac{\sqrt{2} – 1}{2 – 1} = \sqrt{2} – 1 \] \[ a – \frac{1}{a} = (\sqrt{2} + 1) – (\sqrt{2} – 1) \] \[ = 2

Find the Value of 1/x Find the value of \( \frac{1}{x} \) \[ x = \sqrt{2} – 1 \] Solution: \[ \frac{1}{x} = \frac{1}{\sqrt{2} – 1} \times \frac{\sqrt{2} + 1}{\sqrt{2} + 1} \] \[ = \frac{\sqrt{2} + 1}{2 – 1} \] \[ = \sqrt{2} + 1 \] Next Question / Full Exercise

Find x and y Find the values of \(x\) and \(y\) \[ \frac{\sqrt{3} – 1}{\sqrt{3} + 1} = x + y\sqrt{3} \] Solution: \[ \frac{\sqrt{3} – 1}{\sqrt{3} + 1} \times \frac{\sqrt{3} – 1}{\sqrt{3} – 1} \] \[ = \frac{(\sqrt{3} – 1)^2}{3 – 1} \] \[ = \frac{3 – 2\sqrt{3} + 1}{2} \] \[ = \frac{4

Find the Rationalisation Factor Find the rationalisation factor \[ 7 – 3\sqrt{5} \] Solution: \[ \text{Rationalisation factor of } (a – b\sqrt{c}) = (a + b\sqrt{c}) \] \[ \therefore \text{Rationalisation factor of } (7 – 3\sqrt{5}) = 7 + 3\sqrt{5} \] Next Question / Full Exercise

Find the Reciprocal Find the reciprocal \[ \text{Reciprocal of } (5 + \sqrt{2}) = \frac{1}{5 + \sqrt{2}} \] Solution: \[ \frac{1}{5 + \sqrt{2}} \times \frac{5 – \sqrt{2}}{5 – \sqrt{2}} \] \[ = \frac{5 – \sqrt{2}}{5^2 – (\sqrt{2})^2} \] \[ = \frac{5 – \sqrt{2}}{25 – 2} \] \[ = \frac{5 – \sqrt{2}}{23} \] Next Question /