Multiply $ \dfrac{2}{3} $ by $ i $ to get $ \dfrac{ 2i }{ 3 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{2}{3} \cdot i & \xlongequal{\text{Step 1}} \frac{2}{3} \cdot \frac{i}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 2 \cdot i }{ 3 \cdot 1 } \xlongequal{\text{Step 3}} \frac{ 2i }{ 3 } \end{aligned} $$

Add $ \dfrac{2i}{3} $ and $ \dfrac{1}{2} $ to get $ \dfrac{ \color{purple}{ 4i+3 } }{ 6 }$.

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}{ 2 }$ and the second by $\color{blue}{ 3 }$.

$$ \begin{aligned} \frac{2i}{3} + \frac{1}{2} & = \frac{ 2i \cdot \color{blue}{ 2 }}{ 3 \cdot \color{blue}{ 2 }} + \frac{ 1 \cdot \color{blue}{ 3 }}{ 2 \cdot \color{blue}{ 3 }} = \\[1ex] &=\frac{ \color{purple}{ 4i } }{ 6 } + \frac{ \color{purple}{ 3 } }{ 6 }=\frac{ \color{purple}{ 4i+3 } }{ 6 } \end{aligned} $$

Divide $1$ by $ \dfrac{4i+3}{6} $ to get $ \dfrac{ 6 }{ 4i+3 } $.

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.

$$ \begin{aligned} \frac{1}{ \frac{\color{blue}{4i+3}}{\color{blue}{6}} } & \xlongequal{\text{Step 1}} 1 \cdot \frac{\color{blue}{6}}{\color{blue}{4i+3}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{1}{\color{red}{1}} \cdot \frac{6}{4i+3} \xlongequal{\text{Step 3}} \frac{ 1 \cdot 6 }{ 1 \cdot \left( 4i+3 \right) } = \\[1ex] & \xlongequal{\text{Step 4}} \frac{ 6 }{ 4i+3 } \end{aligned} $$