Question

Rationalize the denominator $\frac{1}{2\sqrt{5}-\sqrt{3}}$ and hence find the value of $a$ & $b$ in $\frac{1}{2\sqrt{5}-\sqrt{3}}=a\sqrt{5}+b\sqrt{3}$

Answer 1

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

$\Rightarrow \frac{2\sqrt{5}+\sqrt{3}}{20-3}=\frac{2\sqrt{5}+\sqrt{3}}{17}$

$\Rightarrow \frac{2}{17}\sqrt{5}+\frac{1}{17}\sqrt{3}=a\sqrt{5}+b\sqrt{3}$

So $a=\frac{2}{17}$ & $b=\frac{1}{17}$.