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}{ -\frac{ 14 }{ 5 }+\frac{ 39 }{ 5 }i }\, $ is $ \color{blue}{ -\frac{ 14 }{ 5 }-\frac{ 39 }{ 5 }i } $.

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

$$\begin{aligned} \frac{ \frac{ 53 }{ 25 }+\frac{ 121 }{ 100 }i }{ -\frac{ 14 }{ 5 }+\frac{ 39 }{ 5 }i } &= \frac{ \frac{ 53 }{ 25 }+\frac{ 121 }{ 100 }i }{ -\frac{ 14 }{ 5 }+\frac{ 39 }{ 5 }i } \cdot \frac{ \color{blue}{ -\frac{ 14 }{ 5 }-\frac{ 39 }{ 5 }i } }{ \color{blue}{ -\frac{ 14 }{ 5 }-\frac{ 39 }{ 5 }i } } = \\[1 em] &= \frac{ \left( \frac{ 53 }{ 25 }+\frac{ 121 }{ 100 }i \right) \cdot \left( -\frac{ 14 }{ 5 }-\frac{ 39 }{ 5 }i \right) }{ \left( -\frac{ 14 }{ 5 }+\frac{ 39 }{ 5 }i \right) \cdot \left( -\frac{ 14 }{ 5 }-\frac{ 39 }{ 5 }i \right) } \end{aligned} $$

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

$$ \begin{aligned} \left( \frac{ 53 }{ 25 }+\frac{ 121 }{ 100 }i \right) \cdot \left( -\frac{ 14 }{ 5 }-\frac{ 39 }{ 5 }i \right) &= \frac{ 53 }{ 25 } \cdot \left(-\frac{ 14 }{ 5 }\right) + \frac{ 53 }{ 25 } \cdot \left(-\frac{ 39 }{ 5 } \,i \right) + \left( \frac{ 121 }{ 100 } \,i \right) \cdot \left(-\frac{ 14 }{ 5 } \right) + \left( \frac{ 121 }{ 100 } \,i \right) \cdot \left(-\frac{ 39 }{ 5 } \,i \right) = \\[1 em] &= -\frac{ 742 }{ 125 } -\frac{ 2067 }{ 125 } \, i -\frac{ 847 }{ 250 } \, i -\frac{ 4719 }{ 500 } \color{blue}{(-1)} = \\[1 em] &= \frac{ 1751 }{ 500 }-\frac{ 4981 }{ 250 }i\end{aligned} $$ $$ \begin{aligned} \left( -\frac{ 14 }{ 5 }+\frac{ 39 }{ 5 }i \right) \cdot \left( -\frac{ 14 }{ 5 }-\frac{ 39 }{ 5 }i \right) &= -\frac{ 14 }{ 5 } \cdot \left(-\frac{ 14 }{ 5 }\right) -\frac{ 14 }{ 5 } \cdot \left(-\frac{ 39 }{ 5 } \,i \right) + \left( \frac{ 39 }{ 5 } \,i \right) \cdot \left(-\frac{ 14 }{ 5 } \right) + \left( \frac{ 39 }{ 5 } \,i \right) \cdot \left(-\frac{ 39 }{ 5 } \,i \right) = \\[1 em] &= \frac{ 196 }{ 25 } + \frac{ 546 }{ 25 } \, i -\frac{ 546 }{ 25 } \, i -\frac{ 1521 }{ 25 } \color{blue}{(-1)} = \\[1 em] &= \frac{ 1717 }{ 25 }\end{aligned} $$

Step 4: Separate real and imaginary parts:

$$ \frac{ \frac{ 53 }{ 25 }+\frac{ 121 }{ 100 }i }{ -\frac{ 14 }{ 5 }+\frac{ 39 }{ 5 }i } = \frac{ \frac{ 1751 }{ 500 }-\frac{ 4981 }{ 250 }i }{ \frac{ 1717 }{ 25 } } = \frac{ \frac{ 1751 }{ 500 } }{ \frac{ 1717 }{ 25 } } + \frac{ -\frac{ 4981 }{ 250 } }{ \frac{ 1717 }{ 25 } } i= \frac{ 103 }{ 2020 }-\frac{ 293 }{ 1010 }i $$

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