Add $7$ and $ \dfrac{8}{x} $ to get $ \dfrac{ \color{purple}{ 7x+8 } }{ x }$.

Step 1: Write $ 7 $ 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 $\color{blue}{ x }$.

$$ \begin{aligned} 7+ \frac{8}{x} & \xlongequal{\text{Step 1}} \frac{7}{\color{red}{1}} + \frac{8}{x} = \frac{ 7 \cdot \color{blue}{ x }}{ 1 \cdot \color{blue}{ x }} + \frac{ 8 }{ x } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 7x } }{ x } + \frac{ \color{purple}{ 8 } }{ x }=\frac{ \color{purple}{ 7x+8 } }{ x } \end{aligned} $$

Divide $ \dfrac{7x+8}{x} $ by $ 49 $ to get $ \dfrac{ 7x+8 }{ 49x } $.

Step 1: To divide rational expressions, multiply the first fraction by the reciprocal of the second fraction.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{ \frac{7x+8}{x} }{49} & \xlongequal{\text{Step 1}} \frac{7x+8}{x} \cdot \frac{\color{blue}{1}}{\color{blue}{49}} \xlongequal{\text{Step 2}} \frac{ \left( 7x+8 \right) \cdot 1 }{ x \cdot 49 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 7x+8 }{ 49x } \end{aligned} $$

Divide $ \dfrac{7x+8}{49x} $ by $ x $ to get $ \dfrac{ 7x+8 }{ 49x^2 } $.

Step 1: To divide rational expressions, multiply the first fraction by the reciprocal of the second fraction.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{ \frac{7x+8}{49x} }{x} & \xlongequal{\text{Step 1}} \frac{7x+8}{49x} \cdot \frac{\color{blue}{1}}{\color{blue}{x}} \xlongequal{\text{Step 2}} \frac{ \left( 7x+8 \right) \cdot 1 }{ 49x \cdot x } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 7x+8 }{ 49x^2 } \end{aligned} $$

Subtract $ \dfrac{64}{x^3} $ from $ \dfrac{7x+8}{49x^2} $ to get $ \dfrac{ \color{purple}{ 7x^4+8x^3-3136x^2 } }{ 49x^5 }$.

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}{ x^3 }$ and the second by $\color{blue}{ 49x^2 }$.

$$ \begin{aligned} \frac{7x+8}{49x^2} - \frac{64}{x^3} & = \frac{ \left( 7x+8 \right) \cdot \color{blue}{ x^3 }}{ 49x^2 \cdot \color{blue}{ x^3 }} - \frac{ 64 \cdot \color{blue}{ 49x^2 }}{ x^3 \cdot \color{blue}{ 49x^2 }} = \\[1ex] &=\frac{ \color{purple}{ 7x^4+8x^3 } }{ 49x^5 } - \frac{ \color{purple}{ 3136x^2 } }{ 49x^5 }=\frac{ \color{purple}{ 7x^4+8x^3-3136x^2 } }{ 49x^5 } \end{aligned} $$