Question

Find the value of $a$ and $b$ if $\frac{5+2\sqrt{3}}{7+4\sqrt{3}}=a+b\sqrt{3}$

Answer 1

$\to \frac{5+2\sqrt{3}}{7+4\sqrt{3}}=a+b+\sqrt{3}$

Fins $a$ and $b$ value

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

$=\frac{35-20\sqrt{3}+14\sqrt{3}-24}{7^2-(4\sqrt{3}{)}^2}=\frac{11-6\sqrt{3}}{49-48}=11-6\sqrt{3}$