Question

Find the value of $a$ and $b$
When $a+b\sqrt{5}=\frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}$

Answer 1

$a+b\sqrt{5}=\frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}\times \frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}+\sqrt{3}}$

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

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

$a+b\sqrt{5}=4+\left(\sqrt{3}\right)\left(\sqrt{5}\right)$

$\Rightarrow$ by comparing

$a=4;b=\sqrt{3}$