Add $0i$ and $ \dfrac{1}{25+4i} $ to get $ \dfrac{ \color{purple}{ 1 } }{ 4i+25 }$.

Step 1: Write $ 0i $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the first fraction by $\color{blue}{ 4i+25 }$.

$$ \begin{aligned} 0i+ \frac{1}{25+4i} & \xlongequal{\text{Step 1}} \frac{0i}{\color{red}{1}} + \frac{1}{25+4i} = \frac{ 0i \cdot \color{blue}{ \left( 4i+25 \right) }}{ 1 \cdot \color{blue}{ \left( 4i+25 \right) }} + \frac{ 1 }{ 25+4i } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 0i^20i } }{ 4i+25 } + \frac{ \color{purple}{ 1 } }{ 4i+25 }=\frac{ \color{purple}{ 1 } }{ 4i+25 } \end{aligned} $$

Divide $1$ by $ \dfrac{1}{4i+25} $ to get $ 4i+25$.

Step 1: To divide rational expressions, multiply the first fraction by the reciprocal of the second fraction.

Step 2: Write $ 1 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 3: Multiply numerators and denominators.

Step 4: Simplify numerator and denominator.

$$ \begin{aligned} \frac{1}{ \frac{\color{blue}{1}}{\color{blue}{4i+25}} } & \xlongequal{\text{Step 1}} 1 \cdot \frac{\color{blue}{4i+25}}{\color{blue}{1}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{1}{\color{red}{1}} \cdot \frac{4i+25}{1} \xlongequal{\text{Step 3}} \frac{ 1 \cdot \left( 4i+25 \right) }{ 1 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 4}} \frac{ 4i+25 }{ 1 } =4i+25 \end{aligned} $$