$$ 64 j \cdot 2 = 128 j $$

Multiply $6$ by $ \dfrac{j}{2x} $ to get $ \dfrac{ 6j }{ 2x } $.

Step 1: Write $ 6 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

Subtract $ \dfrac{j}{2c} $ from $ 10 $ to get $ \dfrac{ \color{purple}{ 20c-j } }{ 2c }$.

Step 1: Write $ 10 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

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

$$ \begin{aligned} 10- \frac{j}{2c} & \xlongequal{\text{Step 1}} \frac{10}{\color{red}{1}} - \frac{j}{2c} = \frac{ 10 \cdot \color{blue}{ 2c }}{ 1 \cdot \color{blue}{ 2c }} - \frac{ j }{ 2c } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 20c } }{ 2c } - \frac{ \color{purple}{ j } }{ 2c }=\frac{ \color{purple}{ 20c-j } }{ 2c } \end{aligned} $$

Subtract $ \dfrac{6j}{2x} $ from $ 24+128j $ to get $ \dfrac{ \color{purple}{ 256jx-6j+48x } }{ 2x }$.

Step 1: Write $ 24+128j $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

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

$$ \begin{aligned} 24+128j- \frac{6j}{2x} & \xlongequal{\text{Step 1}} \frac{24+128j}{\color{red}{1}} - \frac{6j}{2x} = \frac{ \left( 24+128j \right) \cdot \color{blue}{ 2x }}{ 1 \cdot \color{blue}{ 2x }} - \frac{ 6j }{ 2x } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 48x+256jx } }{ 2x } - \frac{ \color{purple}{ 6j } }{ 2x }=\frac{ \color{purple}{ 256jx-6j+48x } }{ 2x } \end{aligned} $$

$$ 16 j \cdot 2 = 32 j $$

Add $ \dfrac{256jx-6j+48x}{2x} $ and $ \dfrac{16}{x} $ to get $ \dfrac{ \color{purple}{ 256jx^2-6jx+48x^2+32x } }{ 2x^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}{ x }$ and the second by $\color{blue}{ 2x }$.

$$ \begin{aligned} \frac{256jx-6j+48x}{2x} + \frac{16}{x} & = \frac{ \left( 256jx-6j+48x \right) \cdot \color{blue}{ x }}{ 2x \cdot \color{blue}{ x }} + \frac{ 16 \cdot \color{blue}{ 2x }}{ x \cdot \color{blue}{ 2x }} = \\[1ex] &=\frac{ \color{purple}{ 256jx^2-6jx+48x^2 } }{ 2x^2 } + \frac{ \color{purple}{ 32x } }{ 2x^2 }=\frac{ \color{purple}{ 256jx^2-6jx+48x^2+32x } }{ 2x^2 } \end{aligned} $$

Add $ \dfrac{20c-j}{2c} $ and $ 32j $ to get $ \dfrac{ \color{purple}{ 64cj+20c-j } }{ 2c }$.

Step 1: Write $ 32j $ 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 second fraction by $\color{blue}{2c}$.

$$ \begin{aligned} \frac{20c-j}{2c} +32j & \xlongequal{\text{Step 1}} \frac{20c-j}{2c} + \frac{32j}{\color{red}{1}} = \frac{ 20c-j }{ 2c } + \frac{ 32j \cdot \color{blue}{ 2c }}{ 1 \cdot \color{blue}{ 2c }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 20c-j } }{ 2c } + \frac{ \color{purple}{ 64cj } }{ 2c }=\frac{ \color{purple}{ 64cj+20c-j } }{ 2c } \end{aligned} $$

Divide $ \dfrac{256jx^2-6jx+48x^2+32x}{2x^2} $ by $ \dfrac{64cj+20c-j}{2c} $ to get $ \dfrac{ 512cjx^2-12cjx+96cx^2+64cx }{ 128cjx^2+40cx^2-2jx^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.

$$ \begin{aligned} \frac{ \frac{256jx^2-6jx+48x^2+32x}{2x^2} }{ \frac{\color{blue}{64cj+20c-j}}{\color{blue}{2c}} } & \xlongequal{\text{Step 1}} \frac{256jx^2-6jx+48x^2+32x}{2x^2} \cdot \frac{\color{blue}{2c}}{\color{blue}{64cj+20c-j}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \left( 256jx^2-6jx+48x^2+32x \right) \cdot 2c }{ 2x^2 \cdot \left( 64cj+20c-j \right) } \xlongequal{\text{Step 3}} \frac{ 512cjx^2-12cjx+96cx^2+64cx }{ 128cjx^2+40cx^2-2jx^2 } \end{aligned} $$