Subtract $ \dfrac{1}{8} $ from $ 8x $ to get $ \dfrac{ \color{purple}{ 64x-1 } }{ 8 }$.

Step 1: Write $ 8x $ 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 $\color{blue}{ 8 }$.

$$ \begin{aligned} 8x- \frac{1}{8} & \xlongequal{\text{Step 1}} \frac{8x}{\color{red}{1}} - \frac{1}{8} = \frac{ 8x \cdot \color{blue}{ 8 }}{ 1 \cdot \color{blue}{ 8 }} - \frac{ 1 }{ 8 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 64x } }{ 8 } - \frac{ \color{purple}{ 1 } }{ 8 }=\frac{ \color{purple}{ 64x-1 } }{ 8 } \end{aligned} $$

Add $8x$ and $ \dfrac{1}{8} $ to get $ \dfrac{ \color{purple}{ 64x+1 } }{ 8 }$.

Step 1: Write $ 8x $ 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 $\color{blue}{ 8 }$.

$$ \begin{aligned} 8x+ \frac{1}{8} & \xlongequal{\text{Step 1}} \frac{8x}{\color{red}{1}} + \frac{1}{8} = \frac{ 8x \cdot \color{blue}{ 8 }}{ 1 \cdot \color{blue}{ 8 }} + \frac{ 1 }{ 8 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 64x } }{ 8 } + \frac{ \color{purple}{ 1 } }{ 8 }=\frac{ \color{purple}{ 64x+1 } }{ 8 } \end{aligned} $$

Multiply $ \dfrac{64x-1}{8} $ by $ \dfrac{64x+1}{8} $ to get $ \dfrac{4096x^2-1}{64} $.

Step 1: Multiply numerators and denominators.

Step 2: Simplify numerator and denominator.

$$ \begin{aligned} \frac{64x-1}{8} \cdot \frac{64x+1}{8} & \xlongequal{\text{Step 1}} \frac{ \left( 64x-1 \right) \cdot \left( 64x+1 \right) }{ 8 \cdot 8 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ 4096x^2+ \cancel{64x} -\cancel{64x}-1 }{ 64 } = \frac{4096x^2-1}{64} \end{aligned} $$