Question

If $a$ and $b$ are rational number and $\frac{7-4\sqrt{3}}{7+4\sqrt{3}}=a+b\sqrt{3}$, find the value of a and b

Answer 1

$\frac{7-4\sqrt{3}}{7+4\sqrt{3}}$
$=\frac{7-4\sqrt{3}}{7+4\sqrt{3}}\times \frac{7-4\sqrt{3}}{7-4\sqrt{3}}$
$=\frac{(7-4\sqrt{3}{)}^2}{7^2-(4\sqrt{3}{)}^2}$
$=\frac{49-56\sqrt{3}+48}{1}$
$=97-56\sqrt{3}$
Therefore, $a=97,b=-56$