Add $ \dfrac{z}{z^2} $ and $ 9z $ to get $ \dfrac{ \color{purple}{ 9z^3+z } }{ z^2 }$.

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

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