Question

Express $E=\frac{1}{\sqrt{5}+\sqrt{3}-\sqrt{8}}$ with a rational denominator.

Answer 1

The denominator is a trinomial surd, an expression having all the three terms as surds. We group any two of the three terms, say $\sqrt{5}and\sqrt{3}$.
Thus $\sqrt{5}+\sqrt{3}-\sqrt{8}=(\sqrt{5}+\sqrt{3})-\sqrt{8}$
Consider the product
$\left[\right(\sqrt{5}+\sqrt{3})-\sqrt{8})\left[\right(\sqrt{5}+\sqrt{3})+\sqrt{8})$

$(\sqrt{5}+\sqrt{3}{)}^2-(\sqrt{8}{)}^2$

$5+3+2\sqrt{5}\sqrt{3}-8=2\sqrt{15}$

$\therefore \frac{1}{\sqrt{5}+\sqrt{3}-\sqrt{8}}=\frac{\sqrt{5}+\sqrt{3}+\sqrt{8}}{(\sqrt{5}+\sqrt{3}-\sqrt{8})(\sqrt{5}+\sqrt{3}+\sqrt{8})}$

$=\frac{(\sqrt{5}+\sqrt{3}-\sqrt{8})}{2\sqrt{15}}$

Rationalizing the denominator

$=\frac{\sqrt{5}+\sqrt{3}+\sqrt{8}}{2\sqrt{15}}\left(\frac{\sqrt{15}}{\sqrt{15}}\right)=\frac{5\sqrt{3}+3\sqrt{5}+2\sqrt{30}}{30}$