Question

rationalize the denominator

$\frac{1}{(1 + \sqrt{5} + \sqrt{3}) }$
​

Answer 1

Your answer is in the attachment.

What is rationalisation?

The rewriting of any number in the form of rational numbers is known as rationalisation.

What are real numbers?

The numbers which are in the form of p/q where both p and q are integers and q ≠ 0 are equal numbers.

Answer 2

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

$\mathrm{Multiplying\;and\;Dividing\;the\;above\;fraction\;with\;1 + \sqrt{5} - \sqrt{3}}$

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

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

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

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

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

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

$\mathrm{Multiplying\;and\;Dividing\;the\;above\;fraction\;with\;3 - 2\sqrt{5}}$

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

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

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

$\implies \dfrac{3 + \sqrt{5} - (2)(5) - 3\sqrt{3} + (2\sqrt{5\times 3})}{9 - (4)(5)}}$

$\implies \dfrac{3 + \sqrt{5} - 10 - 3\sqrt{3} + 2\sqrt{15}}{9 - 20}}$

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

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