Educational

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 /

Find the Value of a Find the value of \(a\) \[ \frac{6}{3\sqrt{2} – 2\sqrt{3}} = 3\sqrt{2} – a\sqrt{3} \] Solution: \[ \frac{6}{3\sqrt{2} – 2\sqrt{3}} \times \frac{3\sqrt{2} + 2\sqrt{3}}{3\sqrt{2} + 2\sqrt{3}} \] \[ = \frac{6(3\sqrt{2} + 2\sqrt{3})}{(3\sqrt{2})^2 – (2\sqrt{3})^2} \] \[ = \frac{6(3\sqrt{2} + 2\sqrt{3})}{18 – 12} \] \[ = \frac{6(3\sqrt{2} + 2\sqrt{3})}{6} \] \[ =

Find the Value Find the value \[ (2 + \sqrt{3})(2 – \sqrt{3}) \] Solution: \[ (2 + \sqrt{3})(2 – \sqrt{3}) = 2^2 – (\sqrt{3})^2 \] \[ = 4 – 3 = 1 \] Next Question / Full Exercise

Find the Value Find the value \[ a = 5 + 2\sqrt{6}, \quad b = \frac{1}{a} \] \[ b = \frac{1}{5 + 2\sqrt{6}} \times \frac{5 – 2\sqrt{6}}{5 – 2\sqrt{6}} = 5 – 2\sqrt{6} \] \[ a + b = (5 + 2\sqrt{6}) + (5 – 2\sqrt{6}) = 10 \] \[ a^2 + b^2 = (a

Find the Value Find the value \[ a = \frac{3 + \sqrt{5}}{2} \] \[ \frac{1}{a} = \frac{2}{3 + \sqrt{5}} \times \frac{3 – \sqrt{5}}{3 – \sqrt{5}} = \frac{3 – \sqrt{5}}{2} \] \[ a + \frac{1}{a} = \frac{3 + \sqrt{5}}{2} + \frac{3 – \sqrt{5}}{2} = 3 \] \[ a^2 + \frac{1}{a^2} = (a + \frac{1}{a})^2 – 2

Find the Value Find the value \[ x = \frac{\sqrt{3} + 1}{2} \] \[ x^2 = \frac{(\sqrt{3} + 1)^2}{4} = \frac{3 + 2\sqrt{3} + 1}{4} = \frac{4 + 2\sqrt{3}}{4} = 1 + \frac{\sqrt{3}}{2} \] \[ x^3 = x \cdot x^2 = \frac{\sqrt{3} + 1}{2} \left(1 + \frac{\sqrt{3}}{2}\right) \] \[ = \frac{(\sqrt{3} + 1)(2 + \sqrt{3})}{4}

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

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