Question

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

Answer 1

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

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

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

$\frac{25+12+20\sqrt{3}}{25-12}=a+b\sqrt{3}$

$\frac{37+20\sqrt{3}}{13}=a+b\sqrt{3}$

$\frac{37}{13}+\frac{20}{13}\sqrt{3}=a+b\sqrt{3}$

On comparing, we get,

$a=\frac{37}{13}$ and $b=\frac{20}{13}$