Find the multiplicative inverse of each of the following:

Given: 2 + 5i

To find: Multiplicative inverse

We know that,

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

$=\frac{1}{Z}$

Putting z = 2 + 5i

So, Multiplicative inverse of $2+5 \mathrm{i}=\frac{1}{2+5 \mathrm{i}}$

Now, rationalizing by multiply and divide by the conjugate of (2+5i)

$=\frac{1}{2+5 i} \times \frac{2-5 i}{2-5 i}$

$=\frac{2-5 i}{(2+5 i)(2-5 i)}$

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

$=\frac{2-5 i}{(2)^{2}-(5 i)^{2}}$

$=\frac{2-5 i}{4-25 i^{2}}$

$=\frac{2-5 i}{4-25(-1)}\left[\because i^{2}=-1\right]$

$=\frac{2-5 i}{4+25}$

$=\frac{2-5 i}{29}$

$=\frac{2}{29}-\frac{5}{29} i$

Hence, Multiplicative Inverse of $(2+5 \mathrm{i})$ is $\frac{2}{29}-\frac{5}{29} \mathrm{i}$