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

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{1}{2} \cdot z & \xlongequal{\text{Step 1}} \frac{1}{2} \cdot \frac{z}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 1 \cdot z }{ 2 \cdot 1 } \xlongequal{\text{Step 3}} \frac{ z }{ 2 } \end{aligned} $$

Multiply $ \dfrac{1}{6} $ by $ z^2 $ to get $ \dfrac{ z^2 }{ 6 } $.

Step 1: Write $ z^2 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{1}{6} \cdot z^2 & \xlongequal{\text{Step 1}} \frac{1}{6} \cdot \frac{z^2}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 1 \cdot z^2 }{ 6 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ z^2 }{ 6 } \end{aligned} $$

Multiply $ \dfrac{1}{24} $ by $ z^3 $ to get $ \dfrac{ z^3 }{ 24 } $.

Step 1: Write $ z^3 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{1}{24} \cdot z^3 & \xlongequal{\text{Step 1}} \frac{1}{24} \cdot \frac{z^3}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 1 \cdot z^3 }{ 24 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ z^3 }{ 24 } \end{aligned} $$

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

To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the first fraction by $\color{blue}{ 3 }$.

$$ \begin{aligned} \frac{z}{2} + \frac{z^2}{6} & = \frac{ z \cdot \color{blue}{ 3 }}{ 2 \cdot \color{blue}{ 3 }} + \frac{ z^2 }{ 6 } = \\[1ex] &=\frac{ \color{purple}{ 3z } }{ 6 } + \frac{ \color{purple}{ z^2 } }{ 6 }=\frac{ \color{purple}{ z^2+3z } }{ 6 } \end{aligned} $$

Multiply $ \dfrac{1}{24} $ by $ z^3 $ to get $ \dfrac{ z^3 }{ 24 } $.

Step 1: Write $ z^3 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{1}{24} \cdot z^3 & \xlongequal{\text{Step 1}} \frac{1}{24} \cdot \frac{z^3}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 1 \cdot z^3 }{ 24 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ z^3 }{ 24 } \end{aligned} $$

Add $ \dfrac{z^2+3z}{6} $ and $ \dfrac{z^3}{24} $ to get $ \dfrac{ \color{purple}{ z^3+4z^2+12z } }{ 24 }$.

To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the first fraction by $\color{blue}{ 4 }$.

$$ \begin{aligned} \frac{z^2+3z}{6} + \frac{z^3}{24} & = \frac{ \left( z^2+3z \right) \cdot \color{blue}{ 4 }}{ 6 \cdot \color{blue}{ 4 }} + \frac{ z^3 }{ 24 } = \\[1ex] &=\frac{ \color{purple}{ 4z^2+12z } }{ 24 } + \frac{ \color{purple}{ z^3 } }{ 24 }=\frac{ \color{purple}{ z^3+4z^2+12z } }{ 24 } \end{aligned} $$