Divide $ \dfrac{x^4}{9} $ by $ 75 $ to get $ \dfrac{ x^4 }{ 675 } $.

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.

$$ \begin{aligned} \frac{ \frac{x^4}{9} }{75} & \xlongequal{\text{Step 1}} \frac{x^4}{9} \cdot \frac{\color{blue}{1}}{\color{blue}{75}} \xlongequal{\text{Step 2}} \frac{ x^4 \cdot 1 }{ 9 \cdot 75 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ x^4 }{ 675 } \end{aligned} $$

Multiply $25z^2$ by $ \dfrac{x^4}{675} $ to get $ \dfrac{ 25x^4z^2 }{ 675 } $.

Step 1: Write $ 25z^2 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 25z^2 \cdot \frac{x^4}{675} & \xlongequal{\text{Step 1}} \frac{25z^2}{\color{red}{1}} \cdot \frac{x^4}{675} \xlongequal{\text{Step 2}} \frac{ 25z^2 \cdot x^4 }{ 1 \cdot 675 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 25x^4z^2 }{ 675 } \end{aligned} $$

Multiply $ \dfrac{25x^4z^2}{675} $ by $ x^2 $ to get $ \dfrac{ 25x^6z^2 }{ 675 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{25x^4z^2}{675} \cdot x^2 & \xlongequal{\text{Step 1}} \frac{25x^4z^2}{675} \cdot \frac{x^2}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 25x^4z^2 \cdot x^2 }{ 675 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 25x^6z^2 }{ 675 } \end{aligned} $$

Multiply $ \dfrac{25x^6z^2}{675} $ by $ \dfrac{z^5}{21} $ to get $ \dfrac{ 25x^6z^7 }{ 14175 } $.

Step 1: Multiply numerators and denominators.

Step 2: Simplify numerator and denominator.

$$ \begin{aligned} \frac{25x^6z^2}{675} \cdot \frac{z^5}{21} & \xlongequal{\text{Step 1}} \frac{ 25x^6z^2 \cdot z^5 }{ 675 \cdot 21 } \xlongequal{\text{Step 2}} \frac{ 25x^6z^7 }{ 14175 } \end{aligned} $$