Multiply $abc$ by $ \dfrac{i}{d} $ to get $ \dfrac{ abci }{ d } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} abc \cdot \frac{i}{d} & \xlongequal{\text{Step 1}} \frac{abc}{\color{red}{1}} \cdot \frac{i}{d} \xlongequal{\text{Step 2}} \frac{ abc \cdot i }{ 1 \cdot d } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ abci }{ d } \end{aligned} $$

Add $ \dfrac{abci}{d} $ and $ f $ to get $ \dfrac{ \color{purple}{ abci+df } }{ d }$.

Step 1: Write $ f $ 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 $\color{blue}{d}$.

$$ \begin{aligned} \frac{abci}{d} +f & \xlongequal{\text{Step 1}} \frac{abci}{d} + \frac{f}{\color{red}{1}} = \frac{ abci }{ d } + \frac{ f \cdot \color{blue}{ d }}{ 1 \cdot \color{blue}{ d }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ abci } }{ d } + \frac{ \color{purple}{ df } }{ d }=\frac{ \color{purple}{ abci+df } }{ d } \end{aligned} $$

Multiply $abci$ by $ \dfrac{abci+df}{d} $ to get $ \dfrac{ a^2b^2c^2i^2+abcdfi }{ d } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} abci \cdot \frac{abci+df}{d} & \xlongequal{\text{Step 1}} \frac{abci}{\color{red}{1}} \cdot \frac{abci+df}{d} \xlongequal{\text{Step 2}} \frac{ abci \cdot \left( abci+df \right) }{ 1 \cdot d } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ a^2b^2c^2i^2+abcdfi }{ d } \end{aligned} $$

Divide $1$ by $ \dfrac{a^2b^2c^2i^2+abcdfi}{d} $ to get $ \dfrac{ d }{ a^2b^2c^2i^2+abcdfi } $.

Step 1: To divide rational expressions, multiply the first fraction by the reciprocal of the second fraction.

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

Step 3: Multiply numerators and denominators.

Step 4: Simplify numerator and denominator.

$$ \begin{aligned} \frac{1}{ \frac{\color{blue}{a^2b^2c^2i^2+abcdfi}}{\color{blue}{d}} } & \xlongequal{\text{Step 1}} 1 \cdot \frac{\color{blue}{d}}{\color{blue}{a^2b^2c^2i^2+abcdfi}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{1}{\color{red}{1}} \cdot \frac{d}{a^2b^2c^2i^2+abcdfi} \xlongequal{\text{Step 3}} \frac{ 1 \cdot d }{ 1 \cdot \left( a^2b^2c^2i^2+abcdfi \right) } = \\[1ex] & \xlongequal{\text{Step 4}} \frac{ d }{ a^2b^2c^2i^2+abcdfi } \end{aligned} $$