Multiply $ \dfrac{1}{4} $ by $ i $ to get $ \dfrac{ i }{ 4 } $.

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.

$$ \begin{aligned} \frac{1}{4} \cdot i & \xlongequal{\text{Step 1}} \frac{1}{4} \cdot \frac{i}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 1 \cdot i }{ 4 \cdot 1 } \xlongequal{\text{Step 3}} \frac{ i }{ 4 } \end{aligned} $$

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 $ \dfrac{-1}{6} $ and $ \dfrac{i}{4} $ to get $ \dfrac{ \color{purple}{ 3i-2 } }{ 12 }$.

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 $\color{blue}{ 3 }$.

$$ \begin{aligned} \frac{-1}{6} + \frac{i}{4} & = \frac{ \left( -1 \right) \cdot \color{blue}{ 2 }}{ 6 \cdot \color{blue}{ 2 }} + \frac{ i \cdot \color{blue}{ 3 }}{ 4 \cdot \color{blue}{ 3 }} = \\[1ex] &=\frac{ \color{purple}{ -2 } }{ 12 } + \frac{ \color{purple}{ 3i } }{ 12 }=\frac{ \color{purple}{ 3i-2 } }{ 12 } \end{aligned} $$

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.

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

Subtract $ \dfrac{i}{2} $ from $ \dfrac{3i-2}{12} $ to get $ \dfrac{ \color{purple}{ -3i-2 } }{ 12 }$.

To subtract raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the second fraction by $\color{blue}{6}$.

$$ \begin{aligned} \frac{3i-2}{12} - \frac{i}{2} & = \frac{ 3i-2 }{ 12 } - \frac{ i \cdot \color{blue}{ 6 }}{ 2 \cdot \color{blue}{ 6 }} = \\[1ex] &=\frac{ \color{purple}{ 3i-2 } }{ 12 } - \frac{ \color{purple}{ 6i } }{ 12 }=\frac{ \color{purple}{ -3i-2 } }{ 12 } \end{aligned} $$

Divide $-2-3i$ by $ \dfrac{-3i-2}{12} $ to get $ 12$.

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

Step 2: Write $ -2-3i $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 3: Cancel $ \color{blue}{ -2-3i } $ in first and second fraction.

Step 4: Multiply numerators and denominators.

$$ \begin{aligned} \frac{-2-3i}{ \frac{\color{blue}{-3i-2}}{\color{blue}{12}} } & \xlongequal{\text{Step 1}} -2-3i \cdot \frac{\color{blue}{12}}{\color{blue}{-3i-2}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{-2-3i}{\color{red}{1}} \cdot \frac{12}{-3i-2} \xlongequal{\text{Step 3}} \frac{\color{blue}{1}}{1} \cdot \frac{12}{\color{blue}{1}} = \\[1ex] &= \frac{12}{1} =12 \end{aligned} $$