Educational

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}

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