Question

If$\frac{5+2\sqrt{3}}{7+4\sqrt{3}}=a+b\sqrt{3}$ then find 'a' and 'b'.

Answer 1

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

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

$=\frac{35-20\sqrt{3}+14\sqrt{13}-24}{49-12\times 3}$

$=\frac{11-6\sqrt{3}}{49-36}$

$=\frac{11-6\sqrt{3}}{13}$

$\therefore a=\frac{11}{13}$ and $b=\frac{-6}{13}$