Question

Rationalize the denominator and simplify:
$\frac{5+2\sqrt{3}}{7+4\sqrt{3}}$.

Answer 1

Rationalising the denominator,

$\frac{5+2\sqrt{3}}{7+4\sqrt{3}}$ $=\frac{5+2\sqrt{3}}{7+4\sqrt{3}}\times \frac{7-4\sqrt{3}}{7-4\sqrt{3}}$

$=\frac{(5+2\sqrt{3})(7-4\sqrt{3})}{49-48}=(5+2\sqrt{3})(7-4\sqrt{3})$

$=35-20\sqrt{3}+14\sqrt{3}-24=11-6\sqrt{3}$.