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}{ 21-25i }\, $ is $ \color{blue}{ 21+25i } $.

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

$$\begin{aligned} \frac{ \frac{ 877 }{ 200 }-\frac{ 717 }{ 50 }i }{ 21-25i } &= \frac{ \frac{ 877 }{ 200 }-\frac{ 717 }{ 50 }i }{ 21-25i } \cdot \frac{ \color{blue}{ 21+25i } }{ \color{blue}{ 21+25i } } = \\[1 em] &= \frac{ \left( \frac{ 877 }{ 200 }-\frac{ 717 }{ 50 }i \right) \cdot \left( 21+25i \right) }{ \left( 21-25i \right) \cdot \left( 21+25i \right) } \end{aligned} $$

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

$$ \begin{aligned} \left( \frac{ 877 }{ 200 }-\frac{ 717 }{ 50 }i \right) \cdot \left( 21+25i \right) &= \frac{ 877 }{ 200 } \cdot 21 + \frac{ 877 }{ 200 } \cdot \left(25 \,i \right) + \left( -\frac{ 717 }{ 50 } \,i \right) \cdot \left(21 \right) + \left( -\frac{ 717 }{ 50 } \,i \right) \cdot \left(25 \,i \right) = \\[1 em] &= \frac{ 18417 }{ 200 } + \frac{ 877 }{ 8 } \, i -\frac{ 15057 }{ 50 } \, i -\frac{ 717 }{ 2 } \color{blue}{(-1)} = \\[1 em] &= \frac{ 90117 }{ 200 }-\frac{ 38303 }{ 200 }i\end{aligned} $$ $$ \begin{aligned} \left( 21-25i \right) \cdot \left( 21+25i \right) &= 21 \cdot 21 + 21 \cdot \left(25 \,i \right) + \left( -25 \,i \right) \cdot \left(21 \right) + \left( -25 \,i \right) \cdot \left(25 \,i \right) = \\[1 em] &= 441 + 525 \, i -525 \, i -625 \color{blue}{(-1)} = \\[1 em] &= 1066\end{aligned} $$

Step 4: Separate real and imaginary parts:

$$ \frac{ \frac{ 877 }{ 200 }-\frac{ 717 }{ 50 }i }{ 21-25i } = \frac{ \frac{ 90117 }{ 200 }-\frac{ 38303 }{ 200 }i }{ 1066 } = \frac{ \frac{ 90117 }{ 200 } }{ 1066 } + \frac{ -\frac{ 38303 }{ 200 } }{ 1066 } i= \frac{ 90117 }{ 213200 }-\frac{ 38303 }{ 213200 }i $$

Complex Division Calculator