May 2026

Simplify Expression Simplify \[ \sqrt{7 + 2\sqrt{6}} – \sqrt{7 – 2\sqrt{6}} \] Solution: \[ \sqrt{7 + 2\sqrt{6}} = \sqrt{a} + \sqrt{b}, \quad \sqrt{7 – 2\sqrt{6}} = \sqrt{a} – \sqrt{b} \] \[ a + b = 7, \quad ab = 6 \Rightarrow a = 6,\ b = 1 \] \[ \therefore \sqrt{7 + 2\sqrt{6}} = \sqrt{6} […]

Find the Value of A Find the value of \(A\) \[ \sqrt{5 + 2\sqrt{6}} = A + \sqrt{3} \] Solution: \[ (\,A + \sqrt{3}\,)^2 = 5 + 2\sqrt{6} \] \[ A^2 + 3 + 2A\sqrt{3} = 5 + 2\sqrt{6} \] Comparing rational and irrational parts: \[ A^2 + 3 = 5 \Rightarrow A^2 = 2

Find the Value Find the value \[ x = \frac{2}{\sqrt{3} – \sqrt{5}}, \quad y = \frac{2}{\sqrt{3} + \sqrt{5}} \] Solution: \[ x + y = \frac{2}{\sqrt{3} – \sqrt{5}} + \frac{2}{\sqrt{3} + \sqrt{5}} \] \[ = \frac{2(\sqrt{3} + \sqrt{5}) + 2(\sqrt{3} – \sqrt{5})}{(\sqrt{3})^2 – (\sqrt{5})^2} \] \[ = \frac{2\sqrt{3} + 2\sqrt{5} + 2\sqrt{3} – 2\sqrt{5}}{3 –

Find the Value Find the value \[ x = 3 – \sqrt{8} \] Solution: \[ \frac{1}{x} = \frac{1}{3 – \sqrt{8}} \times \frac{3 + \sqrt{8}}{3 + \sqrt{8}} = 3 + \sqrt{8} \] \[ x + \frac{1}{x} = (3 – \sqrt{8}) + (3 + \sqrt{8}) = 6 \] \[ (x – \frac{1}{x})^2 = (x + \frac{1}{x})^2 –

Find the Value Find the value \[ x = \sqrt{6} + \sqrt{5} \] Solution: \[ \frac{1}{x} = \frac{1}{\sqrt{6} + \sqrt{5}} \times \frac{\sqrt{6} – \sqrt{5}}{\sqrt{6} – \sqrt{5}} \] \[ = \sqrt{6} – \sqrt{5} \] \[ x + \frac{1}{x} = (\sqrt{6} + \sqrt{5}) + (\sqrt{6} – \sqrt{5}) = 2\sqrt{6} \] \[ x^2 + \frac{1}{x^2} – 2 =

Find the Value Find the value \[ a = 5 + 2\sqrt{6} \] Solution: \[ \frac{1}{a} = \frac{1}{5 + 2\sqrt{6}} \times \frac{5 – 2\sqrt{6}}{5 – 2\sqrt{6}} \] \[ = \frac{5 – 2\sqrt{6}}{25 – 24} = 5 – 2\sqrt{6} \] \[ a + \frac{1}{a} = (5 + 2\sqrt{6}) + (5 – 2\sqrt{6}) \] \[ = 10

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

Rationalise the Denominator Find the denominator after rationalisation \[ \frac{7}{3\sqrt{3} – 2\sqrt{2}} \] Solution: \[ \frac{7}{3\sqrt{3} – 2\sqrt{2}} \times \frac{3\sqrt{3} + 2\sqrt{2}}{3\sqrt{3} + 2\sqrt{2}} \] \[ = \frac{7(3\sqrt{3} + 2\sqrt{2})}{(3\sqrt{3})^2 – (2\sqrt{2})^2} \] \[ = \frac{7(3\sqrt{3} + 2\sqrt{2})}{27 – 8} \] \[ = \frac{7(3\sqrt{3} + 2\sqrt{2})}{19} \] \[ \therefore \text{Denominator} = 19 \] Next Question

Find A and B Find the values of \(A\) and \(B\) \[ \frac{1}{\sqrt{9} – \sqrt{8}} = A + B\sqrt{2} \] Solution: \[ \sqrt{9} = 3,\quad \sqrt{8} = 2\sqrt{2} \] \[ \frac{1}{3 – 2\sqrt{2}} \times \frac{3 + 2\sqrt{2}}{3 + 2\sqrt{2}} \] \[ = \frac{3 + 2\sqrt{2}}{9 – 8} \] \[ = 3 + 2\sqrt{2} \] Comparing

Rationalise the Denominator Rationalise the denominator \[ \frac{1}{\sqrt{7} + 2} \] Solution: \[ \frac{1}{\sqrt{7} + 2} \times \frac{\sqrt{7} – 2}{\sqrt{7} – 2} \] \[ = \frac{\sqrt{7} – 2}{7 – 4} \] \[ = \frac{\sqrt{7} – 2}{3} \] Next Question / Full Exercise