Multiply $3$ by $ \dfrac{x}{6} $ to get $ \dfrac{ 3x }{ 6 } $.

Step 1: Write $ 3 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 3 \cdot \frac{x}{6} & \xlongequal{\text{Step 1}} \frac{3}{\color{red}{1}} \cdot \frac{x}{6} \xlongequal{\text{Step 2}} \frac{ 3 \cdot x }{ 1 \cdot 6 } \xlongequal{\text{Step 3}} \frac{ 3x }{ 6 } \end{aligned} $$

Multiply $4$ by $ \dfrac{y}{3} $ to get $ \dfrac{ 4y }{ 3 } $.

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.

$$ \begin{aligned} 4 \cdot \frac{y}{3} & \xlongequal{\text{Step 1}} \frac{4}{\color{red}{1}} \cdot \frac{y}{3} \xlongequal{\text{Step 2}} \frac{ 4 \cdot y }{ 1 \cdot 3 } \xlongequal{\text{Step 3}} \frac{ 4y }{ 3 } \end{aligned} $$

Multiply $4$ by $ \dfrac{x}{5} $ to get $ \dfrac{ 4x }{ 5 } $.

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.

$$ \begin{aligned} 4 \cdot \frac{x}{5} & \xlongequal{\text{Step 1}} \frac{4}{\color{red}{1}} \cdot \frac{x}{5} \xlongequal{\text{Step 2}} \frac{ 4 \cdot x }{ 1 \cdot 5 } \xlongequal{\text{Step 3}} \frac{ 4x }{ 5 } \end{aligned} $$

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

Step 1: Write $ 3 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

Multiply $3$ by $ \dfrac{x}{6} $ to get $ \dfrac{ 3x }{ 6 } $.

Step 1: Write $ 3 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 3 \cdot \frac{x}{6} & \xlongequal{\text{Step 1}} \frac{3}{\color{red}{1}} \cdot \frac{x}{6} \xlongequal{\text{Step 2}} \frac{ 3 \cdot x }{ 1 \cdot 6 } \xlongequal{\text{Step 3}} \frac{ 3x }{ 6 } \end{aligned} $$

Multiply $ \dfrac{4y}{3} $ by $ x $ to get $ \dfrac{ 4xy }{ 3 } $.

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.

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

Multiply $ \dfrac{4x}{5} $ by $ x^2 $ to get $ \dfrac{ 4x^3 }{ 5 } $.

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.

$$ \begin{aligned} \frac{4x}{5} \cdot x^2 & \xlongequal{\text{Step 1}} \frac{4x}{5} \cdot \frac{x^2}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 4x \cdot x^2 }{ 5 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 4x^3 }{ 5 } \end{aligned} $$

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

Step 1: Write $ 3 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

Multiply $3$ by $ \dfrac{x}{6} $ to get $ \dfrac{ 3x }{ 6 } $.

Step 1: Write $ 3 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 3 \cdot \frac{x}{6} & \xlongequal{\text{Step 1}} \frac{3}{\color{red}{1}} \cdot \frac{x}{6} \xlongequal{\text{Step 2}} \frac{ 3 \cdot x }{ 1 \cdot 6 } \xlongequal{\text{Step 3}} \frac{ 3x }{ 6 } \end{aligned} $$

Multiply $ \dfrac{4xy}{3} $ by $ 4 $ to get $ \dfrac{ 16xy }{ 3 } $.

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.

$$ \begin{aligned} \frac{4xy}{3} \cdot 4 & \xlongequal{\text{Step 1}} \frac{4xy}{3} \cdot \frac{4}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 4xy \cdot 4 }{ 3 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 16xy }{ 3 } \end{aligned} $$

Add $ \dfrac{4x^3}{5} $ and $ \dfrac{3y}{x} $ to get $ \dfrac{ \color{purple}{ 4x^4+15y } }{ 5x }$.

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}{ x }$ and the second by $\color{blue}{ 5 }$.

$$ \begin{aligned} \frac{4x^3}{5} + \frac{3y}{x} & = \frac{ 4x^3 \cdot \color{blue}{ x }}{ 5 \cdot \color{blue}{ x }} + \frac{ 3y \cdot \color{blue}{ 5 }}{ x \cdot \color{blue}{ 5 }} = \\[1ex] &=\frac{ \color{purple}{ 4x^4 } }{ 5x } + \frac{ \color{purple}{ 15y } }{ 5x }=\frac{ \color{purple}{ 4x^4+15y } }{ 5x } \end{aligned} $$

Subtract $ \dfrac{16xy}{3} $ from $ \dfrac{3x}{6} $ to get $ \dfrac{ \color{purple}{ -32xy+3x } }{ 6 }$.

To subtract raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the second fraction by $\color{blue}{2}$.

$$ \begin{aligned} \frac{3x}{6} - \frac{16xy}{3} & = \frac{ 3x }{ 6 } - \frac{ 16xy \cdot \color{blue}{ 2 }}{ 3 \cdot \color{blue}{ 2 }} = \\[1ex] &=\frac{ \color{purple}{ 3x } }{ 6 } - \frac{ \color{purple}{ 32xy } }{ 6 }=\frac{ \color{purple}{ -32xy+3x } }{ 6 } \end{aligned} $$

Add $ \dfrac{4x^3}{5} $ and $ \dfrac{3y}{x} $ to get $ \dfrac{ \color{purple}{ 4x^4+15y } }{ 5x }$.

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}{ x }$ and the second by $\color{blue}{ 5 }$.

$$ \begin{aligned} \frac{4x^3}{5} + \frac{3y}{x} & = \frac{ 4x^3 \cdot \color{blue}{ x }}{ 5 \cdot \color{blue}{ x }} + \frac{ 3y \cdot \color{blue}{ 5 }}{ x \cdot \color{blue}{ 5 }} = \\[1ex] &=\frac{ \color{purple}{ 4x^4 } }{ 5x } + \frac{ \color{purple}{ 15y } }{ 5x }=\frac{ \color{purple}{ 4x^4+15y } }{ 5x } \end{aligned} $$

Multiply $ \dfrac{-32xy+3x}{6} $ by $ \dfrac{4x^4+15y}{5x} $ to get $ \dfrac{-128x^5y+12x^5-480xy^2+45xy}{30x} $.

Step 1: Multiply numerators and denominators.

Step 2: Simplify numerator and denominator.

$$ \begin{aligned} \frac{-32xy+3x}{6} \cdot \frac{4x^4+15y}{5x} & \xlongequal{\text{Step 1}} \frac{ \left( -32xy+3x \right) \cdot \left( 4x^4+15y \right) }{ 6 \cdot 5x } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ -128x^5y-480xy^2+12x^5+45xy }{ 30x } = \frac{-128x^5y+12x^5-480xy^2+45xy}{30x} \end{aligned} $$