Multiply $ \dfrac{1}{8} $ by $ x^3 $ to get $ \dfrac{ x^3 }{ 8 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{1}{8} \cdot x^3 & \xlongequal{\text{Step 1}} \frac{1}{8} \cdot \frac{x^3}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 1 \cdot x^3 }{ 8 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ x^3 }{ 8 } \end{aligned} $$

Subtract $ \dfrac{x^3}{8} $ from $ 8 $ to get $ \dfrac{ \color{purple}{ -x^3+64 } }{ 8 }$.

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 first fraction by $\color{blue}{ 8 }$.

$$ \begin{aligned} 8- \frac{x^3}{8} & \xlongequal{\text{Step 1}} \frac{8}{\color{red}{1}} - \frac{x^3}{8} = \frac{ 8 \cdot \color{blue}{ 8 }}{ 1 \cdot \color{blue}{ 8 }} - \frac{ x^3 }{ 8 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 64 } }{ 8 } - \frac{ \color{purple}{ x^3 } }{ 8 }=\frac{ \color{purple}{ -x^3+64 } }{ 8 } \end{aligned} $$