Question

If $\frac{3}{4\sqrt{5}-\sqrt{3}}+\frac{2}{4\sqrt{5}+\sqrt{3}}=a\sqrt{5}+b\sqrt{3}$, then find the values of $a$ and $b$.

Answer 1

$\frac{3}{4\sqrt{5}-\sqrt{3}}+\frac{2}{4\sqrt{5}+\sqrt{3}}$

$=\frac{3(4\sqrt{5}+\sqrt{3})+2(4\sqrt{5}-\sqrt{3})}{(4\sqrt{5}{)}^2-3^2}$

$=\frac{20\sqrt{5}+\sqrt{3}}{71}$

$=\frac{20}{71}\sqrt{5}+\frac{1}{71}\sqrt{3}=a\sqrt{5}+b\sqrt{3}$ [Given]

Comparing both sides we get,

$a=\frac{20}{71},b=\frac{1}{71}$