Add $9$ and $ \dfrac{i}{2} $ to get $ \dfrac{ \color{purple}{ i+18 } }{ 2 }$.

Step 1: Write $ 9 $ 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 $\color{blue}{ 2 }$.

$$ \begin{aligned} 9+ \frac{i}{2} & \xlongequal{\text{Step 1}} \frac{9}{\color{red}{1}} + \frac{i}{2} = \frac{ 9 \cdot \color{blue}{ 2 }}{ 1 \cdot \color{blue}{ 2 }} + \frac{ i }{ 2 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 18 } }{ 2 } + \frac{ \color{purple}{ i } }{ 2 }=\frac{ \color{purple}{ i+18 } }{ 2 } \end{aligned} $$

Multiply $7$ by $ \dfrac{i}{3} $ to get $ \dfrac{ 7i }{ 3 } $.

Step 1: Write $ 7 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 7 \cdot \frac{i}{3} & \xlongequal{\text{Step 1}} \frac{7}{\color{red}{1}} \cdot \frac{i}{3} \xlongequal{\text{Step 2}} \frac{ 7 \cdot i }{ 1 \cdot 3 } \xlongequal{\text{Step 3}} \frac{ 7i }{ 3 } \end{aligned} $$

Add $ \dfrac{1}{9} $ and $ 3i $ to get $ \dfrac{ \color{purple}{ 27i+1 } }{ 9 }$.

Step 1: Write $ 3i $ 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 $\color{blue}{9}$.

$$ \begin{aligned} \frac{1}{9} +3i & \xlongequal{\text{Step 1}} \frac{1}{9} + \frac{3i}{\color{red}{1}} = \frac{ 1 }{ 9 } + \frac{ 3i \cdot \color{blue}{ 9 }}{ 1 \cdot \color{blue}{ 9 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 1 } }{ 9 } + \frac{ \color{purple}{ 27i } }{ 9 }=\frac{ \color{purple}{ 27i+1 } }{ 9 } \end{aligned} $$

Add $ \dfrac{i+18}{2} $ and $ \dfrac{7i}{3} $ to get $ \dfrac{ \color{purple}{ 17i+54 } }{ 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}{ 3 }$ and the second by $\color{blue}{ 2 }$.

$$ \begin{aligned} \frac{i+18}{2} + \frac{7i}{3} & = \frac{ \left( i+18 \right) \cdot \color{blue}{ 3 }}{ 2 \cdot \color{blue}{ 3 }} + \frac{ 7i \cdot \color{blue}{ 2 }}{ 3 \cdot \color{blue}{ 2 }} = \\[1ex] &=\frac{ \color{purple}{ 3i+54 } }{ 6 } + \frac{ \color{purple}{ 14i } }{ 6 }=\frac{ \color{purple}{ 17i+54 } }{ 6 } \end{aligned} $$

Add $ \dfrac{1}{9} $ and $ 3i $ to get $ \dfrac{ \color{purple}{ 27i+1 } }{ 9 }$.

Step 1: Write $ 3i $ 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 $\color{blue}{9}$.

$$ \begin{aligned} \frac{1}{9} +3i & \xlongequal{\text{Step 1}} \frac{1}{9} + \frac{3i}{\color{red}{1}} = \frac{ 1 }{ 9 } + \frac{ 3i \cdot \color{blue}{ 9 }}{ 1 \cdot \color{blue}{ 9 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 1 } }{ 9 } + \frac{ \color{purple}{ 27i } }{ 9 }=\frac{ \color{purple}{ 27i+1 } }{ 9 } \end{aligned} $$

Add $ \dfrac{17i+54}{6} $ and $ \dfrac{27i+1}{9} $ to get $ \dfrac{ \color{purple}{ 105i+164 } }{ 18 }$.

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

$$ \begin{aligned} \frac{17i+54}{6} + \frac{27i+1}{9} & = \frac{ \left( 17i+54 \right) \cdot \color{blue}{ 3 }}{ 6 \cdot \color{blue}{ 3 }} + \frac{ \left( 27i+1 \right) \cdot \color{blue}{ 2 }}{ 9 \cdot \color{blue}{ 2 }} = \\[1ex] &=\frac{ \color{purple}{ 51i+162 } }{ 18 } + \frac{ \color{purple}{ 54i+2 } }{ 18 }=\frac{ \color{purple}{ 105i+164 } }{ 18 } \end{aligned} $$