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}{ 50-5i }\, $ is $ \color{blue}{ 50+5i } $.

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

$$\begin{aligned} \frac{ 500-1250i }{ 50-5i } &= \frac{ 500-1250i }{ 50-5i } \cdot \frac{ \color{blue}{ 50+5i } }{ \color{blue}{ 50+5i } } = \\[1 em] &= \frac{ \left( 500-1250i \right) \cdot \left( 50+5i \right) }{ \left( 50-5i \right) \cdot \left( 50+5i \right) } \end{aligned} $$

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

$$ \begin{aligned} \left( 500-1250i \right) \cdot \left( 50+5i \right) &= 500 \cdot 50 + 500 \cdot \left(5 \,i \right) + \left( -1250 \,i \right) \cdot \left(50 \right) + \left( -1250 \,i \right) \cdot \left(5 \,i \right) = \\[1 em] &= 25000 + 2500 \, i -62500 \, i -6250 \color{blue}{(-1)} = \\[1 em] &= 31250-60000i\end{aligned} $$ $$ \begin{aligned} \left( 50-5i \right) \cdot \left( 50+5i \right) &= 50 \cdot 50 + 50 \cdot \left(5 \,i \right) + \left( -5 \,i \right) \cdot \left(50 \right) + \left( -5 \,i \right) \cdot \left(5 \,i \right) = \\[1 em] &= 2500 + 250 \, i -250 \, i -25 \color{blue}{(-1)} = \\[1 em] &= 2525\end{aligned} $$

Step 4: Separate real and imaginary parts:

$$ \frac{ 500-1250i }{ 50-5i } = \frac{ 31250-60000i }{ 2525 } = \frac{ 31250 }{ 2525 } + \frac{ -60000 }{ 2525 } i= \frac{ 1250 }{ 101 }-\frac{ 2400 }{ 101 }i $$

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