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{ 1 }{ 5 }-\frac{ 3759 }{ 25 }i }\, $ is $ \color{blue}{ \frac{ 1 }{ 5 }+\frac{ 3759 }{ 25 }i } $.

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

$$\begin{aligned} \frac{ 1 }{ \frac{ 1 }{ 5 }-\frac{ 3759 }{ 25 }i } &= \frac{ 1 }{ \frac{ 1 }{ 5 }-\frac{ 3759 }{ 25 }i } \cdot \frac{ \color{blue}{ \frac{ 1 }{ 5 }+\frac{ 3759 }{ 25 }i } }{ \color{blue}{ \frac{ 1 }{ 5 }+\frac{ 3759 }{ 25 }i } } = \\[1 em] &= \frac{ 1 \cdot \left( \frac{ 1 }{ 5 }+\frac{ 3759 }{ 25 }i \right) }{ \left( \frac{ 1 }{ 5 }-\frac{ 3759 }{ 25 }i \right) \cdot \left( \frac{ 1 }{ 5 }+\frac{ 3759 }{ 25 }i \right) } \end{aligned} $$

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

$$ \begin{aligned} 1 \cdot \left( \frac{ 1 }{ 5 }+\frac{ 3759 }{ 25 }i \right) &= 1 \cdot \frac{ 1 }{ 5 } + 1 \cdot \left(\frac{ 3759 }{ 25 } \,i \right) = \\[1 em] &= \frac{ 1 }{ 5 } + \frac{ 3759 }{ 25 } \, i = \\[1 em] &= \frac{ 1 }{ 5 }+\frac{ 3759 }{ 25 }i\end{aligned} $$ $$ \begin{aligned} \left( \frac{ 1 }{ 5 }-\frac{ 3759 }{ 25 }i \right) \cdot \left( \frac{ 1 }{ 5 }+\frac{ 3759 }{ 25 }i \right) &= \frac{ 1 }{ 5 } \cdot \frac{ 1 }{ 5 } + \frac{ 1 }{ 5 } \cdot \left(\frac{ 3759 }{ 25 } \,i \right) + \left( -\frac{ 3759 }{ 25 } \,i \right) \cdot \left(\frac{ 1 }{ 5 } \right) + \left( -\frac{ 3759 }{ 25 } \,i \right) \cdot \left(\frac{ 3759 }{ 25 } \,i \right) = \\[1 em] &= \frac{ 1 }{ 25 } + \frac{ 3759 }{ 125 } \, i -\frac{ 3759 }{ 125 } \, i -\frac{ 14130081 }{ 625 } \color{blue}{(-1)} = \\[1 em] &= \frac{ 14130106 }{ 625 }\end{aligned} $$

Step 4: Separate real and imaginary parts:

$$ \frac{ 1 }{ \frac{ 1 }{ 5 }-\frac{ 3759 }{ 25 }i } = \frac{ \frac{ 1 }{ 5 }+\frac{ 3759 }{ 25 }i }{ \frac{ 14130106 }{ 625 } } = \frac{ \frac{ 1 }{ 5 } }{ \frac{ 14130106 }{ 625 } } + \frac{ \frac{ 3759 }{ 25 } }{ \frac{ 14130106 }{ 625 } } i= \frac{ 125 }{ 14130106 }+\frac{ 93975 }{ 14130106 }i $$

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