# How do you rationalize the denominator and simplify 1/ (sqrt 3 - sqrt 5)?

May 13, 2016

$- \frac{1}{2} \left(\sqrt{3} + \sqrt{5}\right)$

#### Explanation:

What we have to do here is to multiply the numerator and denominator by the$\textcolor{b l u e}{\text{ conjugate"" of the denominator}}$

The conjugate here is $\sqrt{3} + \sqrt{5}$

Note that the radicals remain unchanged ,while the 'sign' changes.

If $\sqrt{3} - \sqrt{5} \text{ then conjugate} = \sqrt{3} + \sqrt{5}$

$\Rightarrow \frac{1}{\sqrt{3} - \sqrt{5}} = \frac{1 \left(\sqrt{3} + \sqrt{5}\right)}{\left(\sqrt{3} - \sqrt{5}\right) \left(\sqrt{3} + \sqrt{5}\right)}$

Consider the denominator.

$\left(\sqrt{3} - \sqrt{5}\right) \left(\sqrt{3} + \sqrt{5}\right) \text{ and expand using FOIL}$

$= {\left(\sqrt{3}\right)}^{2} + \cancel{\sqrt{5} . \sqrt{3}} - \cancel{\sqrt{5} . \sqrt{3}} - {\left(\sqrt{5}\right)}^{2}$

$= 3 - 5 = - 2 \text{ which is rational}$

$\Rightarrow \frac{1}{\sqrt{3} - \sqrt{5}} = \frac{\sqrt{3} + \sqrt{5}}{- 2} = - \frac{1}{2} \left(\sqrt{3} + \sqrt{5}\right)$