Find the multiplicative inverse of each of the following:

Given: $(1-i \sqrt{3})$

To find: Multiplicative inverse

We know that,

Multiplicative Inverse of $z=z^{-1}$

$=\frac{1}{Z}$

Putting $z=1-i \sqrt{3}$

So, Multiplicative inverse of $1-\mathrm{i} \sqrt{3}=\frac{1}{1-\mathrm{i} \sqrt{3}}$

Now, rationalizing by multiply and divide by the conjugate of $(1-\mathrm{i} \sqrt{3})$

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

$=\frac{1+i \sqrt{3}}{(1-i \sqrt{3})(1+i \sqrt{3})}$

Using $(a-b)(a+b)=\left(a^{2}-b^{2}\right)$

$=\frac{1+i \sqrt{3}}{(1)^{2}-(i \sqrt{3})^{2}}$

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

$=\frac{1+i \sqrt{3}}{1-3(-1)}\left[\because i^{2}=-1\right]$

$=\frac{1+i \sqrt{3}}{1+3}$

$=\frac{1+i \sqrt{3}}{4}$

$=\frac{1}{4}+\frac{\sqrt{3}}{4} i$

Hence, Multiplicative Inverse of $(1-\mathrm{i} \sqrt{3})$ is $\frac{1}{4}+\frac{\sqrt{3}}{4} \mathrm{i}$