May 2026

Rationalising Factor Find the simplest rationalising factor \[ 2\sqrt{5} – \sqrt{3} \] Solution: \[ \text{Rationalising factor of } (a – b) = (a + b) \] \[ \therefore \text{Rationalising factor of } (2\sqrt{5} – \sqrt{3}) = 2\sqrt{5} + \sqrt{3} \] \[ (2\sqrt{5} – \sqrt{3})(2\sqrt{5} + \sqrt{3}) = 20 – 3 = 17 \ (\text{rational}) \] […]

Rationalising Factor Find the simplest rationalising factor \[ \sqrt{3} + \sqrt{5} \] Solution: \[ \text{Rationalising factor of } (a + b) = (a – b) \] \[ \therefore \text{Rationalising factor of } (\sqrt{3} + \sqrt{5}) = \sqrt{3} – \sqrt{5} \] \[ (\sqrt{3} + \sqrt{5})(\sqrt{3} – \sqrt{5}) = 3 – 5 = -2 \ (\text{rational}) \]

Find a and b Find the values of \(a\) and \(b\) \[ \frac{\sqrt{3} – 1}{\sqrt{3} + 1} = a – b\sqrt{3} \] 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 Value Find the value \[ x = \sqrt{5} + 2 \] Solution: \[ \frac{1}{x} = \frac{1}{\sqrt{5} + 2} \times \frac{\sqrt{5} – 2}{\sqrt{5} – 2} \] \[ = \frac{\sqrt{5} – 2}{5 – 4} = \sqrt{5} – 2 \] \[ x – \frac{1}{x} = (\sqrt{5} + 2) – (\sqrt{5} – 2) \] \[ = 4

Rationalisation Factor Find the rationalisation factor \[ 2 + \sqrt{3} \] Solution: \[ \text{Rationalisation factor of } (2 + \sqrt{3}) = (2 – \sqrt{3}) \] \[ \because (2 + \sqrt{3})(2 – \sqrt{3}) = 4 – 3 = 1 \] \[ \therefore \text{Answer: } 2 – \sqrt{3} \] Next Question / Full Exercise

Rationalisation Factor Find the rationalisation factor \[ \sqrt{3} \] Solution: \[ \text{Rationalisation factor of } \sqrt{3} = \frac{1}{\sqrt{3}} \] \[ \because \sqrt{3} \times \frac{1}{\sqrt{3}} = 1 \ (\text{a rational number}) \] \[ \therefore \text{Correct Answer: } \frac{1}{\sqrt{3}} \ (\text{Option b}) \] Next Question / Full Exercise

Simplify Surd Simplify \[ \sqrt[5]{6} \times \sqrt[5]{6} \] Solution: \[ \sqrt[5]{6} \times \sqrt[5]{6} = \sqrt[5]{6 \times 6} \] \[ = \sqrt[5]{36} \] \[ \therefore \text{Correct Answer: } \sqrt[5]{36} \ (\text{Option a}) \] Next Question / Full Exercise

Simplify Surd Simplify \[ \sqrt{10} \times \sqrt{15} \] Solution: \[ \sqrt{10} \times \sqrt{15} = \sqrt{150} \] \[ = \sqrt{25 \times 6} \] \[ = 5\sqrt{6} \] \[ \therefore \text{Correct Answer: } 5\sqrt{6} \ (\text{Option a}) \] Next Question / Full Exercise

Find the Value of x Find the value of \(x\) \[ \sqrt{13 – x\sqrt{10}} = \sqrt{8} + \sqrt{5} \] Solution: \[ \sqrt{8} = 2\sqrt{2} \Rightarrow \sqrt{8} + \sqrt{5} = 2\sqrt{2} + \sqrt{5} \] \[ (\,2\sqrt{2} + \sqrt{5}\,)^2 = 13 – x\sqrt{10} \] \[ = 8 + 5 + 4\sqrt{10} = 13 + 4\sqrt{10} \] \[

Find the Value Find the value \[ x = \frac{2}{\sqrt{10} – \sqrt{8}}, \quad y = \frac{2}{\sqrt{10} + 2\sqrt{2}} \] Solution: \[ x = \frac{2}{\sqrt{10} – 2\sqrt{2}} \times \frac{\sqrt{10} + 2\sqrt{2}}{\sqrt{10} + 2\sqrt{2}} = \frac{2(\sqrt{10} + 2\sqrt{2})}{10 – 8} \] \[ = \sqrt{10} + 2\sqrt{2} \] \[ y = \frac{2}{\sqrt{10} + 2\sqrt{2}} \times \frac{\sqrt{10} – 2\sqrt{2}}{\sqrt{10}