Question

Find the multiplicative inverse of: $\frac{2-\sqrt{3}}{2+\sqrt{3}}$

Answer 1

We know that the multiplicative inverse of a number $x\ne 0$ is $\frac{1}{x}$, which when multiplied by $x$ yeilds the multiplicative identity $1$ that is

$x\times \frac{1}{x}=1$

Now, since

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

Hence, the multiplicative inverse of $\frac{2-\sqrt{3}}{2+\sqrt{3}}$ is $\frac{2+\sqrt{3}}{2-\sqrt{3}}$.