Multiply $27$ by $ \dfrac{z^6}{9} $ to get $ \dfrac{ 27z^6 }{ 9 } $.

Step 1: Write $ 27 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 27 \cdot \frac{z^6}{9} & \xlongequal{\text{Step 1}} \frac{27}{\color{red}{1}} \cdot \frac{z^6}{9} \xlongequal{\text{Step 2}} \frac{ 27 \cdot z^6 }{ 1 \cdot 9 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 27z^6 }{ 9 } \end{aligned} $$

Multiply $ \dfrac{27z^6}{9} $ by $ z^4 $ to get $ \dfrac{ 27z^{10} }{ 9 } $.

Step 1: Write $ z^4 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{27z^6}{9} \cdot z^4 & \xlongequal{\text{Step 1}} \frac{27z^6}{9} \cdot \frac{z^4}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 27z^6 \cdot z^4 }{ 9 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 27z^{10} }{ 9 } \end{aligned} $$

Subtract $ \dfrac{27z^{10}}{9} $ from $ 18z^{11} $ to get $ \dfrac{ \color{purple}{ 162z^{11}-27z^{10} } }{ 9 }$.

Step 1: Write $ 18z^{11} $ 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 first fraction by $\color{blue}{ 9 }$.

$$ \begin{aligned} 18z^{11}- \frac{27z^{10}}{9} & \xlongequal{\text{Step 1}} \frac{18z^{11}}{\color{red}{1}} - \frac{27z^{10}}{9} = \frac{ 18z^{11} \cdot \color{blue}{ 9 }}{ 1 \cdot \color{blue}{ 9 }} - \frac{ 27z^{10} }{ 9 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 162z^{11} } }{ 9 } - \frac{ \color{purple}{ 27z^{10} } }{ 9 }=\frac{ \color{purple}{ 162z^{11}-27z^{10} } }{ 9 } \end{aligned} $$