Add $ \dfrac{1}{2} $ and $ \dfrac{8}{3} $ to get $ \dfrac{ \color{purple}{ 19 } }{ 6 }$.

To add fractions they must have the same denominator.
We can create a common denominator by multiplying the first fraction by $ \color{blue}{ 3 }$ and the second by $\color{blue}{ 2 }$.

$$ \begin{aligned} \frac{1}{2} + \frac{8}{3} & = \frac{ 1 \cdot \color{blue}{ 3 }}{ 2 \cdot \color{blue}{ 3 }} + \frac{ 8 \cdot \color{blue}{ 2 }}{ 3 \cdot \color{blue}{ 2 }} = \\[1ex] &=\frac{ \color{purple}{ 3 } }{ 6 } + \frac{ \color{purple}{ 16 } }{ 6 }=\frac{ \color{purple}{ 19 } }{ 6 } \end{aligned} $$

Add $ \dfrac{2}{x^2} $ and $ \dfrac{x}{4} $ to get $ \dfrac{ \color{purple}{ x^3+8 } }{ 4x^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}{ 4 }$ and the second by $\color{blue}{ x^2 }$.

$$ \begin{aligned} \frac{2}{x^2} + \frac{x}{4} & = \frac{ 2 \cdot \color{blue}{ 4 }}{ x^2 \cdot \color{blue}{ 4 }} + \frac{ x \cdot \color{blue}{ x^2 }}{ 4 \cdot \color{blue}{ x^2 }} = \\[1ex] &=\frac{ \color{purple}{ 8 } }{ 4x^2 } + \frac{ \color{purple}{ x^3 } }{ 4x^2 }=\frac{ \color{purple}{ x^3+8 } }{ 4x^2 } \end{aligned} $$

Divide $ \dfrac{19}{6} $ by $ \dfrac{x^3+8}{4x^2} $ to get $ \dfrac{ 76x^2 }{ 6x^3+48 } $.

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.

$$ \begin{aligned} \frac{ \frac{19}{6} }{ \frac{\color{blue}{x^3+8}}{\color{blue}{4x^2}} } & \xlongequal{\text{Step 1}} \frac{19}{6} \cdot \frac{\color{blue}{4x^2}}{\color{blue}{x^3+8}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ 19 \cdot 4x^2 }{ 6 \cdot \left( x^3+8 \right) } \xlongequal{\text{Step 3}} \frac{ 76x^2 }{ 6x^3+48 } \end{aligned} $$