Multiply $2x$ by $ \dfrac{y}{3} $ to get $ \dfrac{ 2xy }{ 3 } $.

Step 1: Write $ 2x $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

Multiply $8$ by $ \dfrac{y^2}{3} $ to get $ \dfrac{ 8y^2 }{ 3 } $.

Step 1: Write $ 8 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

Multiply $2x$ by $ \dfrac{y}{3} $ to get $ \dfrac{ 2xy }{ 3 } $.

Step 1: Write $ 2x $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

Multiply $ \dfrac{8y^2}{3} $ by $ \dfrac{1}{x} $ to get $ \dfrac{ 8y^2 }{ 3x } $.

Step 1: Multiply numerators and denominators.

Step 2: Simplify numerator and denominator.

$$ \begin{aligned} \frac{8y^2}{3} \cdot \frac{1}{x} \xlongequal{\text{Step 1}} \frac{ 8y^2 \cdot 1 }{ 3 \cdot x } \xlongequal{\text{Step 2}} \frac{ 8y^2 }{ 3x } \end{aligned} $$

Multiply $2x$ by $ \dfrac{y}{3} $ to get $ \dfrac{ 2xy }{ 3 } $.

Step 1: Write $ 2x $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

Add $x$ and $ \dfrac{8y^2}{3x} $ to get $ \dfrac{ \color{purple}{ 3x^2+8y^2 } }{ 3x }$.

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

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

Divide $ \dfrac{2xy}{3} $ by $ \dfrac{3x^2+8y^2}{3x} $ to get $ \dfrac{ 6x^2y }{ 9x^2+24y^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.

$$ \begin{aligned} \frac{ \frac{2xy}{3} }{ \frac{\color{blue}{3x^2+8y^2}}{\color{blue}{3x}} } & \xlongequal{\text{Step 1}} \frac{2xy}{3} \cdot \frac{\color{blue}{3x}}{\color{blue}{3x^2+8y^2}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ 2xy \cdot 3x }{ 3 \cdot \left( 3x^2+8y^2 \right) } \xlongequal{\text{Step 3}} \frac{ 6x^2y }{ 9x^2+24y^2 } \end{aligned} $$