Step 1: Determine the conjugate of the denominator. ( to find the conjugate just change the sign of the imaginary part ).

In this example, the conjugate of $ \color{orangered}{ 14+4i }\, $ is $ \color{blue}{ 14-4i } $.

Step 2: Multiply both the numerator and denominator by the conjugate:

$$\begin{aligned} \frac{ 2+4i }{ 14+4i } &= \frac{ 2+4i }{ 14+4i } \cdot \frac{ \color{blue}{ 14-4i } }{ \color{blue}{ 14-4i } } = \\[1 em] &= \frac{ \left( 2+4i \right) \cdot \left( 14-4i \right) }{ \left( 14+4i \right) \cdot \left( 14-4i \right) } \end{aligned} $$

Step 3: Simplify numerator and denominator (use $\color{blue}{i^2 = -1}$)

$$ \begin{aligned} \left( 2+4i \right) \cdot \left( 14-4i \right) &= 2 \cdot 14 + 2 \cdot \left(-4 \,i \right) + \left( 4 \,i \right) \cdot \left(14 \right) + \left( 4 \,i \right) \cdot \left(-4 \,i \right) = \\[1 em] &= 28 -8 \, i + 56 \, i -16 \color{blue}{(-1)} = \\[1 em] &= 44+48i\end{aligned} $$ $$ \begin{aligned} \left( 14+4i \right) \cdot \left( 14-4i \right) &= 14 \cdot 14 + 14 \cdot \left(-4 \,i \right) + \left( 4 \,i \right) \cdot \left(14 \right) + \left( 4 \,i \right) \cdot \left(-4 \,i \right) = \\[1 em] &= 196 -56 \, i + 56 \, i -16 \color{blue}{(-1)} = \\[1 em] &= 212\end{aligned} $$

Step 4: Separate real and imaginary parts:

$$ \frac{ 2+4i }{ 14+4i } = \frac{ 44+48i }{ 212 } = \frac{ 44 }{ 212 } + \frac{ 48 }{ 212 } i= \frac{ 11 }{ 53 }+\frac{ 12 }{ 53 }i $$

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