Add $x^3$ and $ \dfrac{13}{x^3} $ to get $ \dfrac{ \color{purple}{ x^6+13 } }{ x^3 }$.

Step 1: Write $ x^3 $ 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 $\color{blue}{ x^3 }$.

$$ \begin{aligned} x^3+ \frac{13}{x^3} & \xlongequal{\text{Step 1}} \frac{x^3}{\color{red}{1}} + \frac{13}{x^3} = \frac{ x^3 \cdot \color{blue}{ x^3 }}{ 1 \cdot \color{blue}{ x^3 }} + \frac{ 13 }{ x^3 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ x^6 } }{ x^3 } + \frac{ \color{purple}{ 13 } }{ x^3 }=\frac{ \color{purple}{ x^6+13 } }{ x^3 } \end{aligned} $$

Subtract $64$ from $ \dfrac{x^6+13}{x^3} $ to get $ \dfrac{ \color{purple}{ x^6-64x^3+13 } }{ x^3 }$.

Step 1: Write $ 64 $ 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 $\color{blue}{x^3}$.

$$ \begin{aligned} \frac{x^6+13}{x^3} -64 & \xlongequal{\text{Step 1}} \frac{x^6+13}{x^3} - \frac{64}{\color{red}{1}} = \frac{ x^6+13 }{ x^3 } - \frac{ 64 \cdot \color{blue}{ x^3 }}{ 1 \cdot \color{blue}{ x^3 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ x^6+13 } }{ x^3 } - \frac{ \color{purple}{ 64x^3 } }{ x^3 }=\frac{ \color{purple}{ x^6-64x^3+13 } }{ x^3 } \end{aligned} $$