Question

Rationalise the denominator and simplify ${\displaystyle \frac{\sqrt{5}}{\sqrt{6}+2}-\frac{\sqrt{5}}{\sqrt{6}-2}}$

Answer 1

${\displaystyle \frac{\sqrt{5}}{\sqrt{6}+2}-\frac{\sqrt{5}}{\sqrt{6}-2}=\sqrt{5}(\frac{1}{\sqrt{6}+2}-\frac{1}{\sqrt{6}-2})}$

= ${\displaystyle \sqrt{5}\left[\frac{\sqrt{6}-2-(\sqrt{6}+2)}{(\sqrt{6}+2)(\sqrt{6}-2)}\right]}$

= ${\displaystyle \sqrt{5}\left[\frac{\sqrt{6}-2-\sqrt{6}-2}{{\left(\sqrt{6}\right)}^2-2^2}\right]}$

= ${\displaystyle \sqrt{5}\left(\frac{-4}{6-4}\right)}$

= ${\displaystyle \sqrt{5}\left(\frac{-4}{2}\right)}$

= ${\displaystyle \sqrt{5}\times -2}$

= ${\displaystyle -2\sqrt{5}}$