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}{ 47-13i }\, $ is $ \color{blue}{ 47+13i } $.

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

$$\begin{aligned} \frac{ 53+56i }{ 47-13i } &= \frac{ 53+56i }{ 47-13i } \cdot \frac{ \color{blue}{ 47+13i } }{ \color{blue}{ 47+13i } } = \\[1 em] &= \frac{ \left( 53+56i \right) \cdot \left( 47+13i \right) }{ \left( 47-13i \right) \cdot \left( 47+13i \right) } \end{aligned} $$

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

$$ \begin{aligned} \left( 53+56i \right) \cdot \left( 47+13i \right) &= 53 \cdot 47 + 53 \cdot \left(13 \,i \right) + \left( 56 \,i \right) \cdot \left(47 \right) + \left( 56 \,i \right) \cdot \left(13 \,i \right) = \\[1 em] &= 2491 + 689 \, i + 2632 \, i + 728 \color{blue}{(-1)} = \\[1 em] &= 1763+3321i\end{aligned} $$ $$ \begin{aligned} \left( 47-13i \right) \cdot \left( 47+13i \right) &= 47 \cdot 47 + 47 \cdot \left(13 \,i \right) + \left( -13 \,i \right) \cdot \left(47 \right) + \left( -13 \,i \right) \cdot \left(13 \,i \right) = \\[1 em] &= 2209 + 611 \, i -611 \, i -169 \color{blue}{(-1)} = \\[1 em] &= 2378\end{aligned} $$

Step 4: Separate real and imaginary parts:

$$ \frac{ 53+56i }{ 47-13i } = \frac{ 1763+3321i }{ 2378 } = \frac{ 1763 }{ 2378 } + \frac{ 3321 }{ 2378 } i= \frac{ 43 }{ 58 }+\frac{ 81 }{ 58 }i $$

Complex Division Calculator