Question

If 'a' and 'b' are rational numbers, then find the values of 'a' and 'b' given that $\frac{2+5\sqrt{3}}{2-\sqrt{3}}=a+b\sqrt{3}$

Answer 1

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

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

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

$\frac{4+2\sqrt{3}+10\sqrt{3}+15}{4-3}=a+b\sqrt{3}$

$19+12\sqrt{3}=a+b\sqrt{3}$

Comparing we get, $a=19,b=12$