Multiply $ \dfrac{1}{2} $ by $ i $ to get $ \dfrac{ i }{ 2 } $.

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.

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

Add $1+2i$ and $ \dfrac{i}{2} $ to get $ \dfrac{ \color{purple}{ 5i+2 } }{ 2 }$.

Step 1: Write $ 1+2i $ 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 $\color{blue}{ 2 }$.

$$ \begin{aligned} 1+2i+ \frac{i}{2} & \xlongequal{\text{Step 1}} \frac{1+2i}{\color{red}{1}} + \frac{i}{2} = \frac{ \left( 1+2i \right) \cdot \color{blue}{ 2 }}{ 1 \cdot \color{blue}{ 2 }} + \frac{ i }{ 2 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 2+4i } }{ 2 } + \frac{ \color{purple}{ i } }{ 2 }=\frac{ \color{purple}{ 5i+2 } }{ 2 } \end{aligned} $$

Divide $ \dfrac{1}{2+3i} $ by $ \dfrac{5i+2}{2} $ to get $ \dfrac{2}{15i^2+16i+4} $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{ \frac{1}{2+3i} }{ \frac{\color{blue}{5i+2}}{\color{blue}{2}} } & \xlongequal{\text{Step 1}} \frac{1}{2+3i} \cdot \frac{\color{blue}{2}}{\color{blue}{5i+2}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ 1 \cdot 2 }{ \left( 2+3i \right) \cdot \left( 5i+2 \right) } \xlongequal{\text{Step 3}} \frac{ 2 }{ 10i+4+15i^2+6i } = \\[1ex] &= \frac{2}{15i^2+16i+4} \end{aligned} $$

$$ 15i^2 = 15 \cdot (-1) = -15 $$

Simplify denominator

$$ \color{blue}{-15} +16i+ \color{blue}{4} = 16i \color{blue}{-11} $$