If $x=\sqrt{13}+2 \sqrt{3}$, find the value of $x-\frac{1}{x}$.
$x=\sqrt{13}+2 \sqrt{3} \quad \ldots .(1)$
$\Rightarrow \frac{1}{x}=\frac{1}{\sqrt{13}+2 \sqrt{3}}$
$\Rightarrow \frac{1}{x}=\frac{1}{\sqrt{13}+2 \sqrt{3}} \times \frac{\sqrt{13}-2 \sqrt{3}}{\sqrt{13}-2 \sqrt{3}}$
$\Rightarrow \frac{1}{x}=\frac{\sqrt{13}-2 \sqrt{3}}{(\sqrt{13})^{2}-(2 \sqrt{3})^{2}}$
$\Rightarrow \frac{1}{x}=\frac{\sqrt{13}-2 \sqrt{3}}{13-12}$
$\Rightarrow \frac{1}{x}=\sqrt{13}-2 \sqrt{3}$ ..........(2)
Subtracting (2) from (1), we get
$x-\frac{1}{x}=(\sqrt{13}+2 \sqrt{3})-(\sqrt{13}-2 \sqrt{3})$
$\Rightarrow x-\frac{1}{x}=\sqrt{13}+2 \sqrt{3}-\sqrt{13}+2 \sqrt{3}$
$\Rightarrow x-\frac{1}{x}=4 \sqrt{3}$
Thus, the value of $x-\frac{1}{x}$ is $4 \sqrt{3}$.
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