Question

If ${}^{\prime }{a}^{\prime }$ and ${}^{\prime }{b}^{\prime }$ 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

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

Lets take LHS: $\frac{\sqrt{5}+\sqrt{3}}{2\sqrt{5}-3\sqrt{3}}$

Multiply numerator and denominator by $2\sqrt{5}+3\sqrt{3}$, we get

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

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

$=\frac{10+2\sqrt{15}+3\sqrt{15}+9}{20-27}$

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

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

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

So, by compering both, we get

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