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

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

Multiply $4$ by $ \dfrac{y^2}{x^2} $ to get $ \dfrac{ 4y^2 }{ x^2 } $.

Step 1: Write $ 4 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 4 \cdot \frac{y^2}{x^2} & \xlongequal{\text{Step 1}} \frac{4}{\color{red}{1}} \cdot \frac{y^2}{x^2} \xlongequal{\text{Step 2}} \frac{ 4 \cdot y^2 }{ 1 \cdot x^2 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 4y^2 }{ x^2 } \end{aligned} $$

Multiply $ \dfrac{8x^3}{y^3} $ by $ \dfrac{4y^2}{x^2} $ to get $ \dfrac{ 32x^3y^2 }{ x^2y^3 } $.

Step 1: Multiply numerators and denominators.

Step 2: Simplify numerator and denominator.

$$ \begin{aligned} \frac{8x^3}{y^3} \cdot \frac{4y^2}{x^2} & \xlongequal{\text{Step 1}} \frac{ 8x^3 \cdot 4y^2 }{ y^3 \cdot x^2 } \xlongequal{\text{Step 2}} \frac{ 32x^3y^2 }{ x^2y^3 } \end{aligned} $$