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{ 221 }{ 50 }-\frac{ 141 }{ 10 }i }\, $ is $ \color{blue}{ \frac{ 221 }{ 50 }+\frac{ 141 }{ 10 }i } $.

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

$$\begin{aligned} \frac{ \frac{ 553 }{ 100 }-\frac{ 1457 }{ 100 }i }{ \frac{ 221 }{ 50 }-\frac{ 141 }{ 10 }i } &= \frac{ \frac{ 553 }{ 100 }-\frac{ 1457 }{ 100 }i }{ \frac{ 221 }{ 50 }-\frac{ 141 }{ 10 }i } \cdot \frac{ \color{blue}{ \frac{ 221 }{ 50 }+\frac{ 141 }{ 10 }i } }{ \color{blue}{ \frac{ 221 }{ 50 }+\frac{ 141 }{ 10 }i } } = \\[1 em] &= \frac{ \left( \frac{ 553 }{ 100 }-\frac{ 1457 }{ 100 }i \right) \cdot \left( \frac{ 221 }{ 50 }+\frac{ 141 }{ 10 }i \right) }{ \left( \frac{ 221 }{ 50 }-\frac{ 141 }{ 10 }i \right) \cdot \left( \frac{ 221 }{ 50 }+\frac{ 141 }{ 10 }i \right) } \end{aligned} $$

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

$$ \begin{aligned} \left( \frac{ 553 }{ 100 }-\frac{ 1457 }{ 100 }i \right) \cdot \left( \frac{ 221 }{ 50 }+\frac{ 141 }{ 10 }i \right) &= \frac{ 553 }{ 100 } \cdot \frac{ 221 }{ 50 } + \frac{ 553 }{ 100 } \cdot \left(\frac{ 141 }{ 10 } \,i \right) + \left( -\frac{ 1457 }{ 100 } \,i \right) \cdot \left(\frac{ 221 }{ 50 } \right) + \left( -\frac{ 1457 }{ 100 } \,i \right) \cdot \left(\frac{ 141 }{ 10 } \,i \right) = \\[1 em] &= \frac{ 122213 }{ 5000 } + \frac{ 77973 }{ 1000 } \, i -\frac{ 321997 }{ 5000 } \, i -\frac{ 205437 }{ 1000 } \color{blue}{(-1)} = \\[1 em] &= \frac{ 574699 }{ 2500 }+\frac{ 16967 }{ 1250 }i\end{aligned} $$ $$ \begin{aligned} \left( \frac{ 221 }{ 50 }-\frac{ 141 }{ 10 }i \right) \cdot \left( \frac{ 221 }{ 50 }+\frac{ 141 }{ 10 }i \right) &= \frac{ 221 }{ 50 } \cdot \frac{ 221 }{ 50 } + \frac{ 221 }{ 50 } \cdot \left(\frac{ 141 }{ 10 } \,i \right) + \left( -\frac{ 141 }{ 10 } \,i \right) \cdot \left(\frac{ 221 }{ 50 } \right) + \left( -\frac{ 141 }{ 10 } \,i \right) \cdot \left(\frac{ 141 }{ 10 } \,i \right) = \\[1 em] &= \frac{ 48841 }{ 2500 } + \frac{ 31161 }{ 500 } \, i -\frac{ 31161 }{ 500 } \, i -\frac{ 19881 }{ 100 } \color{blue}{(-1)} = \\[1 em] &= \frac{ 272933 }{ 1250 }\end{aligned} $$

Step 4: Separate real and imaginary parts:

$$ \frac{ \frac{ 553 }{ 100 }-\frac{ 1457 }{ 100 }i }{ \frac{ 221 }{ 50 }-\frac{ 141 }{ 10 }i } = \frac{ \frac{ 574699 }{ 2500 }+\frac{ 16967 }{ 1250 }i }{ \frac{ 272933 }{ 1250 } } = \frac{ \frac{ 574699 }{ 2500 } }{ \frac{ 272933 }{ 1250 } } + \frac{ \frac{ 16967 }{ 1250 } }{ \frac{ 272933 }{ 1250 } } i= \frac{ 574699 }{ 545866 }+\frac{ 16967 }{ 272933 }i $$

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