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}{ 80+\frac{ 3201 }{ 4 }i }\, $ is $ \color{blue}{ 80-\frac{ 3201 }{ 4 }i } $.

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

$$\begin{aligned} \frac{ 1000-5i }{ 80+\frac{ 3201 }{ 4 }i } &= \frac{ 1000-5i }{ 80+\frac{ 3201 }{ 4 }i } \cdot \frac{ \color{blue}{ 80-\frac{ 3201 }{ 4 }i } }{ \color{blue}{ 80-\frac{ 3201 }{ 4 }i } } = \\[1 em] &= \frac{ \left( 1000-5i \right) \cdot \left( 80-\frac{ 3201 }{ 4 }i \right) }{ \left( 80+\frac{ 3201 }{ 4 }i \right) \cdot \left( 80-\frac{ 3201 }{ 4 }i \right) } \end{aligned} $$

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

$$ \begin{aligned} \left( 1000-5i \right) \cdot \left( 80-\frac{ 3201 }{ 4 }i \right) &= 1000 \cdot 80 + 1000 \cdot \left(-\frac{ 3201 }{ 4 } \,i \right) + \left( -5 \,i \right) \cdot \left(80 \right) + \left( -5 \,i \right) \cdot \left(-\frac{ 3201 }{ 4 } \,i \right) = \\[1 em] &= 80000 -800250 \, i -400 \, i + \frac{ 16005 }{ 4 } \color{blue}{(-1)} = \\[1 em] &= \frac{ 303995 }{ 4 }-800650i\end{aligned} $$ $$ \begin{aligned} \left( 80+\frac{ 3201 }{ 4 }i \right) \cdot \left( 80-\frac{ 3201 }{ 4 }i \right) &= 80 \cdot 80 + 80 \cdot \left(-\frac{ 3201 }{ 4 } \,i \right) + \left( \frac{ 3201 }{ 4 } \,i \right) \cdot \left(80 \right) + \left( \frac{ 3201 }{ 4 } \,i \right) \cdot \left(-\frac{ 3201 }{ 4 } \,i \right) = \\[1 em] &= 6400 -64020 \, i + 64020 \, i -\frac{ 10246401 }{ 16 } \color{blue}{(-1)} = \\[1 em] &= \frac{ 10348801 }{ 16 }\end{aligned} $$

Step 4: Separate real and imaginary parts:

$$ \frac{ 1000-5i }{ 80+\frac{ 3201 }{ 4 }i } = \frac{ \frac{ 303995 }{ 4 }-800650i }{ \frac{ 10348801 }{ 16 } } = \frac{ \frac{ 303995 }{ 4 } }{ \frac{ 10348801 }{ 16 } } + \frac{ -800650 }{ \frac{ 10348801 }{ 16 } } i= \frac{ 1215980 }{ 10348801 }-\frac{ 12810400 }{ 10348801 }i $$

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