Add $z$ and $ \dfrac{1}{z} $ to get $ \dfrac{ \color{purple}{ z^2+1 } }{ z }$.

Step 1: Write $ z $ 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 $\color{blue}{ z }$.

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