Question

If $a$ and $b$ are rational numbers, find the value of $a$ and $b$ in each of the following equations.
$\frac{\sqrt{5}+\sqrt{3}}{2\sqrt{5}-3\sqrt{3}}=a-b\sqrt{15}$

Answer 1

$\frac{\sqrt{5}+\sqrt{3}}{2\sqrt{5}-3\sqrt{3}}=a-b\sqrt{15}$

$\frac{\left(\sqrt{5}+\sqrt{3}\right)\left(2\sqrt{5}+3\sqrt{3}\right)}{\left(2\sqrt{5}-3\sqrt{3}\right)\left(2\sqrt{5}+3\sqrt{3}\right)}=a+b\sqrt{15}$

$\frac{2\times 5+3\sqrt{15}+2\sqrt{15}+9}{{\left(2\sqrt{15}\right)}^2-{\left(3\sqrt{3}\right)}^2}=a-b\sqrt{15}$

$\frac{19+5\sqrt{15}}{20-27}=a-b\sqrt{15}$

$\frac{-19}{7}\frac{-5\sqrt{15}}{7}=a-b\sqrt{15}$