Add $ \dfrac{1}{-45i} $ and $ \dfrac{1}{61+38i} $ to get $ \dfrac{ \color{purple}{ -7i+61 } }{ -1710i^2-2745i }$.

To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the first fraction by $ \color{blue}{ 38i+61 }$ and the second by $\color{blue}{ -45i }$.

$$ \begin{aligned} \frac{1}{-45i} + \frac{1}{61+38i} & = \frac{ 1 \cdot \color{blue}{ \left( 38i+61 \right) }}{ \left( -45i \right) \cdot \color{blue}{ \left( 38i+61 \right) }} + \frac{ 1 \cdot \color{blue}{ \left( -45i \right) }}{ \left( 61+38i \right) \cdot \color{blue}{ \left( -45i \right) }} = \\[1ex] &=\frac{ \color{purple}{ 38i+61 } }{ -1710i^2-2745i } + \frac{ \color{purple}{ -45i } }{ -1710i^2-2745i } = \\[1ex] &=\frac{ \color{purple}{ -7i+61 } }{ -1710i^2-2745i } \end{aligned} $$

$$ -1710i^2 = -1710 \cdot (-1) = 1710 $$

Divide $ \, 61-7i \, $ by $ \, 1710-2745i \, $ to get $\,\, \dfrac{549+691i}{46485} $.
( view steps )