Add $z^2-14z$ and $ \dfrac{24}{z^3} $ to get $ \dfrac{ \color{purple}{ z^5-14z^4+24 } }{ z^3 }$.

Step 1: Write $ z^2-14z $ 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 $\color{blue}{ z^3 }$.

$$ \begin{aligned} z^2-14z+ \frac{24}{z^3} & \xlongequal{\text{Step 1}} \frac{z^2-14z}{\color{red}{1}} + \frac{24}{z^3} = \frac{ \left( z^2-14z \right) \cdot \color{blue}{ z^3 }}{ 1 \cdot \color{blue}{ z^3 }} + \frac{ 24 }{ z^3 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ z^5-14z^4 } }{ z^3 } + \frac{ \color{purple}{ 24 } }{ z^3 }=\frac{ \color{purple}{ z^5-14z^4+24 } }{ z^3 } \end{aligned} $$

Subtract $8$ from $ \dfrac{z^5-14z^4+24}{z^3} $ to get $ \dfrac{ \color{purple}{ z^5-14z^4-8z^3+24 } }{ z^3 }$.

Step 1: Write $ 8 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: To subtract raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the second fraction by $\color{blue}{z^3}$.

$$ \begin{aligned} \frac{z^5-14z^4+24}{z^3} -8 & \xlongequal{\text{Step 1}} \frac{z^5-14z^4+24}{z^3} - \frac{8}{\color{red}{1}} = \frac{ z^5-14z^4+24 }{ z^3 } - \frac{ 8 \cdot \color{blue}{ z^3 }}{ 1 \cdot \color{blue}{ z^3 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ z^5-14z^4+24 } }{ z^3 } - \frac{ \color{purple}{ 8z^3 } }{ z^3 }=\frac{ \color{purple}{ z^5-14z^4-8z^3+24 } }{ z^3 } \end{aligned} $$