Multiply $ai$ by $ \dfrac{b}{a} $ to get $ \dfrac{ abi }{ a } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} ai \cdot \frac{b}{a} & \xlongequal{\text{Step 1}} \frac{ai}{\color{red}{1}} \cdot \frac{b}{a} \xlongequal{\text{Step 2}} \frac{ ai \cdot b }{ 1 \cdot a } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ abi }{ a } \end{aligned} $$

Add $ \dfrac{abi}{a} $ and $ bi $ to get $ \dfrac{ \color{purple}{ 2abi } }{ a }$.

Step 1: Write $ bi $ 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 second fraction by $\color{blue}{a}$.

$$ \begin{aligned} \frac{abi}{a} +bi & \xlongequal{\text{Step 1}} \frac{abi}{a} + \frac{bi}{\color{red}{1}} = \frac{ abi }{ a } + \frac{ bi \cdot \color{blue}{ a }}{ 1 \cdot \color{blue}{ a }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ abi } }{ a } + \frac{ \color{purple}{ abi } }{ a }=\frac{ \color{purple}{ 2abi } }{ a } \end{aligned} $$