Find the Value Find the value If \[ x = \frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} – \sqrt{3}} \quad \text{and} \quad y = \frac{\sqrt{5} – \sqrt{3}}{\sqrt{5} + \sqrt{3}}, \] then \[ x + y + xy = \ ? \] Solution: \[ x = \frac{(\sqrt{5} + \sqrt{3})^2}{5 – 3} = \frac{5 + 3 + 2\sqrt{15}}{2} = 4 + […]

Find the Value Find the value \[ x + \sqrt{15} = 4 \] Solution: \[ x = 4 – \sqrt{15} \] \[ \frac{1}{x} = \frac{1}{4 – \sqrt{15}} \times \frac{4 + \sqrt{15}}{4 + \sqrt{15}} \] \[ = \frac{4 + \sqrt{15}}{16 – 15} = 4 + \sqrt{15} \] \[ x + \frac{1}{x} = (4 – \sqrt{15}) +

Find the Value Find the value \[ x = 7 + 4\sqrt{3}, \quad xy = 1 \] Solution: \[ y = \frac{1}{x} = 7 – 4\sqrt{3} \] \[ x + y = (7 + 4\sqrt{3}) + (7 – 4\sqrt{3}) = 14 \] \[ xy = 1 \] \[ x^2 + y^2 = (x + y)^2

Find the Value Find the value \[ x = \frac{2}{3 + \sqrt{7}} \] Solution: \[ x = \frac{2}{3 + \sqrt{7}} \times \frac{3 – \sqrt{7}}{3 – \sqrt{7}} = \frac{2(3 – \sqrt{7})}{9 – 7} \] \[ = 3 – \sqrt{7} \] \[ x – 3 = (3 – \sqrt{7}) – 3 = -\sqrt{7} \] \[ (x –

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