$$ i^5 = i^{4 \cdot 1 + 1} = \left( i^4 \right)^{ 1 } \cdot i^1 = 1^{ 1 } \cdot i = i $$

Multiply $5i$ by $ \dfrac{1}{2} $ to get $ \dfrac{ 5i }{ 2 } $.

Step 1: Write $ 5i $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 5i \cdot \frac{1}{2} & \xlongequal{\text{Step 1}} \frac{5i}{\color{red}{1}} \cdot \frac{1}{2} \xlongequal{\text{Step 2}} \frac{ 5i \cdot 1 }{ 1 \cdot 2 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 5i }{ 2 } \end{aligned} $$

Add $3$ and $ \dfrac{5i}{2} $ to get $ \dfrac{ \color{purple}{ 5i+6 } }{ 2 }$.

Step 1: Write $ 3 $ 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 $\color{blue}{ 2 }$.

$$ \begin{aligned} 3+ \frac{5i}{2} & \xlongequal{\text{Step 1}} \frac{3}{\color{red}{1}} + \frac{5i}{2} = \frac{ 3 \cdot \color{blue}{ 2 }}{ 1 \cdot \color{blue}{ 2 }} + \frac{ 5i }{ 2 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 6 } }{ 2 } + \frac{ \color{purple}{ 5i } }{ 2 }=\frac{ \color{purple}{ 5i+6 } }{ 2 } \end{aligned} $$

Add $ \dfrac{5i+6}{2} $ and $ i $ to get $ \dfrac{ \color{purple}{ 7i+6 } }{ 2 }$.

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

$$ \begin{aligned} \frac{5i+6}{2} +i & \xlongequal{\text{Step 1}} \frac{5i+6}{2} + \frac{i}{\color{red}{1}} = \frac{ 5i+6 }{ 2 } + \frac{ i \cdot \color{blue}{ 2 }}{ 1 \cdot \color{blue}{ 2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 5i+6 } }{ 2 } + \frac{ \color{purple}{ 2i } }{ 2 }=\frac{ \color{purple}{ 7i+6 } }{ 2 } \end{aligned} $$