Question

it's answer is (7-underoot5+3underoot3 -2underroot 15 ) over 11​

Answer 1

$\mathrm{Question :\;\dfrac{1}{1 + \sqrt{5} + \sqrt{3}}}$

$\mathrm{\mathbf{Step - 1} : Multiplying\;and\;Dividing\;with\;1 + \sqrt{5} - \sqrt{3}}$

$\mathrm{\implies \dfrac{1 + \sqrt{5} - \sqrt{3}}{1 + \sqrt{5} - \sqrt{3}} \times \dfrac{1}{1 + \sqrt{5} + \sqrt{3}}}$

$\mathrm{\implies \dfrac{1 + \sqrt{5} - \sqrt{3}}{(1 + \sqrt{5} - \sqrt{3})(1 + \sqrt{5} - \sqrt{3})}}$

$\mathrm{\implies \dfrac{1 + \sqrt{5} - \sqrt{3}}{(1 + \sqrt{5})^2 - (\sqrt{3})^2}}$

$\mathrm{\implies \dfrac{1 + \sqrt{5} - \sqrt{3}}{1 + 5 + 2\sqrt{5} - 3}}$

$\mathrm{\implies \dfrac{1 + \sqrt{5} - \sqrt{3}}{3 + 2\sqrt{5}}}$

$\mathrm{\mathbf{Step - 2} : Multiplying\;and\;Dividing\;with\;3 - 2\sqrt{5}}$

$\mathrm{\implies \dfrac{3 - 2\sqrt{5}}{3 - 2\sqrt{5}} \times \dfrac{1 + \sqrt{5} - \sqrt{3}}{3 + 2\sqrt{5}}}$

$\mathrm{\implies \dfrac{(3 - 2\sqrt{5})(1 + \sqrt{5} - \sqrt{3})}{(3 + 2\sqrt{5})(3 - 2\sqrt{5})}}$

$\mathrm{\implies \dfrac{3 + 3\sqrt{5} - 3\sqrt{3} - 2\sqrt{5} - 10 + 2\sqrt{15}}{(3)^2 - (2\sqrt{5})^2}}$

$\mathrm{\implies \dfrac{-7 + \sqrt{5} - 3\sqrt{3} + 2\sqrt{15}}{9 - 20}}$

$\mathrm{\implies \dfrac{7 - \sqrt{5} + 3\sqrt{3} - 2\sqrt{15}}{11}}$