Multiply $ \dfrac{17}{8} $ by $ i $ to get $ \dfrac{ 17i }{ 8 } $.

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.

$$ \begin{aligned} \frac{17}{8} \cdot i & \xlongequal{\text{Step 1}} \frac{17}{8} \cdot \frac{i}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 17 \cdot i }{ 8 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 17i }{ 8 } \end{aligned} $$

Multiply $ \dfrac{27}{8} $ by $ i $ to get $ \dfrac{ 27i }{ 8 } $.

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.

$$ \begin{aligned} \frac{27}{8} \cdot i & \xlongequal{\text{Step 1}} \frac{27}{8} \cdot \frac{i}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 27 \cdot i }{ 8 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 27i }{ 8 } \end{aligned} $$

Multiply $ \dfrac{17i}{8} $ by $ p $ to get $ \dfrac{ 17ip }{ 8 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{17i}{8} \cdot p & \xlongequal{\text{Step 1}} \frac{17i}{8} \cdot \frac{p}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 17i \cdot p }{ 8 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 17ip }{ 8 } \end{aligned} $$

Multiply $ \dfrac{27i}{8} $ by $ p $ to get $ \dfrac{ 27ip }{ 8 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{27i}{8} \cdot p & \xlongequal{\text{Step 1}} \frac{27i}{8} \cdot \frac{p}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 27i \cdot p }{ 8 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 27ip }{ 8 } \end{aligned} $$

$$ i^2 = -1 $$

$$ i^3 = \color{blue}{i^2} \cdot i = ( \color{blue}{-1}) \cdot i = - \, i $$

Multiply $ \dfrac{17ip}{8} $ by $ -1 $ to get $ \dfrac{ -17ip }{ 8 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{17ip}{8} \cdot -1 & \xlongequal{\text{Step 1}} \frac{17ip}{8} \cdot \frac{-1}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 17ip \cdot \left( -1 \right) }{ 8 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ -17ip }{ 8 } \end{aligned} $$

Multiply $ \dfrac{27ip}{8} $ by $ -i $ to get $ \dfrac{ -27i^2p }{ 8 } $.

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.

$$ \begin{aligned} \frac{27ip}{8} \cdot -i & \xlongequal{\text{Step 1}} \frac{27ip}{8} \cdot \frac{-i}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 27ip \cdot \left( -i \right) }{ 8 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ -27i^2p }{ 8 } \end{aligned} $$

Add $ \dfrac{-17ip}{8} $ and $ \dfrac{-27i^2p}{8} $ to get $ \dfrac{-27i^2p-17ip}{8} $.

To add expressions with the same denominators, we add the numerators and write the result over the common denominator.

$$ \begin{aligned} \frac{-17ip}{8} + \frac{-27i^2p}{8} & = \frac{-17ip}{\color{blue}{8}} + \frac{-27i^2p}{\color{blue}{8}} = \\[1ex] &=\frac{ -17ip + \left( -27i^2p \right) }{ \color{blue}{ 8 }}= \frac{-27i^2p-17ip}{8} \end{aligned} $$

Divide $ \dfrac{-27i^2p-17ip}{8} $ by $ 2 $ to get $ \dfrac{ -27i^2p-17ip }{ 16 } $.

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.

$$ \begin{aligned} \frac{ \frac{-27i^2p-17ip}{8} }{2} & \xlongequal{\text{Step 1}} \frac{-27i^2p-17ip}{8} \cdot \frac{\color{blue}{1}}{\color{blue}{2}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \left( -27i^2p-17ip \right) \cdot 1 }{ 8 \cdot 2 } \xlongequal{\text{Step 3}} \frac{ -27i^2p-17ip }{ 16 } \end{aligned} $$

Add $ \dfrac{-27i^2p-17ip}{16} $ and $ 2i $ to get $ \dfrac{ \color{purple}{ -27i^2p-17ip+32i } }{ 16 }$.

Step 1: Write $ 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 second fraction by $\color{blue}{16}$.

$$ \begin{aligned} \frac{-27i^2p-17ip}{16} +2i & \xlongequal{\text{Step 1}} \frac{-27i^2p-17ip}{16} + \frac{2i}{\color{red}{1}} = \frac{ -27i^2p-17ip }{ 16 } + \frac{ 2i \cdot \color{blue}{ 16 }}{ 1 \cdot \color{blue}{ 16 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ -27i^2p-17ip } }{ 16 } + \frac{ \color{purple}{ 32i } }{ 16 }=\frac{ \color{purple}{ -27i^2p-17ip+32i } }{ 16 } \end{aligned} $$