Question

If $a=6+2\sqrt{3}$, then find $a-\frac{1}{a}$

Answer 1

$a=6+2\sqrt{3}$

$\Rightarrow \frac{1}{a}=\frac{1}{6+2\sqrt{3}}$

$\Rightarrow \frac{1}{a}=\frac{1}{6+2\sqrt{3}}\times \frac{6-2\sqrt{3}}{6-2\sqrt{3}}$

$\Rightarrow \frac{1}{a}=\frac{6-2\sqrt{3}}{6^2-{\left(2\sqrt{3}\right)}^2}$

$\Rightarrow \frac{1}{a}=\frac{6-2\sqrt{3}}{24}=\frac{3-\sqrt{3}}{12}$

Therefore,

$a-\frac{1}{a}=(6+2\sqrt{3})-\left(\frac{3-\sqrt{3}}{12}\right)$

$\Rightarrow a-\frac{1}{a}=6+2\sqrt{3}-\frac{3-\sqrt{3}}{12}$

$\Rightarrow a-\frac{1}{a}=\frac{72+24\sqrt{3}-3+\sqrt{3}}{12}=\frac{69+25\sqrt{3}}{12}$