Question

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

Answer 1

Find the value of $a$ and $b$ in $\frac{5+2\sqrt{3}}{7+4\sqrt{3}}=a-b\sqrt{3}$by rationalizing the denominator

Consider the L.H.S of the given equation

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

Rationalize the denominator

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

$=\frac{\left(35–20\sqrt{3}+14\sqrt{3}–24\right)}{49-48}$

$=11-6\sqrt{3}$

Compare L.H.S with R.H.S

$\Rightarrow 11-6\sqrt{3}=a-b\sqrt{3}$

$\Rightarrow a=11,b=6$

Hence the answer is $a=11,b=6$