Multiply $ \dfrac{1}{7} $ by $ x $ to get $ \dfrac{ x }{ 7 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{1}{7} \cdot x & \xlongequal{\text{Step 1}} \frac{1}{7} \cdot \frac{x}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 1 \cdot x }{ 7 \cdot 1 } \xlongequal{\text{Step 3}} \frac{ x }{ 7 } \end{aligned} $$

Multiply $ \dfrac{x}{7} $ by $ y^4 $ to get $ \dfrac{ xy^4 }{ 7 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{x}{7} \cdot y^4 & \xlongequal{\text{Step 1}} \frac{x}{7} \cdot \frac{y^4}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ x \cdot y^4 }{ 7 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ xy^4 }{ 7 } \end{aligned} $$

Add $8x^3y^2-7xy^2$ and $ \dfrac{xy^4}{7} $ to get $ \dfrac{ \color{purple}{ 56x^3y^2+xy^4-49xy^2 } }{ 7 }$.

Step 1: Write $ 8x^3y^2-7xy^2 $ 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 $\color{blue}{ 7 }$.

$$ \begin{aligned} 8x^3y^2-7xy^2+ \frac{xy^4}{7} & \xlongequal{\text{Step 1}} \frac{8x^3y^2-7xy^2}{\color{red}{1}} + \frac{xy^4}{7} = \frac{ \left( 8x^3y^2-7xy^2 \right) \cdot \color{blue}{ 7 }}{ 1 \cdot \color{blue}{ 7 }} + \frac{ xy^4 }{ 7 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 56x^3y^2-49xy^2 } }{ 7 } + \frac{ \color{purple}{ xy^4 } }{ 7 }=\frac{ \color{purple}{ 56x^3y^2+xy^4-49xy^2 } }{ 7 } \end{aligned} $$