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}{ 1-3i }\, $ is $ \color{blue}{ 1+3i } $.

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

$$\begin{aligned} \frac{ 20+10i }{ 1-3i } &= \frac{ 20+10i }{ 1-3i } \cdot \frac{ \color{blue}{ 1+3i } }{ \color{blue}{ 1+3i } } = \\[1 em] &= \frac{ \left( 20+10i \right) \cdot \left( 1+3i \right) }{ \left( 1-3i \right) \cdot \left( 1+3i \right) } \end{aligned} $$

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

$$ \begin{aligned} \left( 20+10i \right) \cdot \left( 1+3i \right) &= 20 \cdot 1 + 20 \cdot \left(3 \,i \right) + \left( 10 \,i \right) \cdot \left(1 \right) + \left( 10 \,i \right) \cdot \left(3 \,i \right) = \\[1 em] &= 20 + 60 \, i + 10 \, i + 30 \color{blue}{(-1)} = \\[1 em] &= -10+70i\end{aligned} $$ $$ \begin{aligned} \left( 1-3i \right) \cdot \left( 1+3i \right) &= 1 \cdot 1 + 1 \cdot \left(3 \,i \right) + \left( -3 \,i \right) \cdot \left(1 \right) + \left( -3 \,i \right) \cdot \left(3 \,i \right) = \\[1 em] &= 1 + 3 \, i -3 \, i -9 \color{blue}{(-1)} = \\[1 em] &= 10\end{aligned} $$

Step 4: Separate real and imaginary parts:

$$ \frac{ 20+10i }{ 1-3i } = \frac{ -10+70i }{ 10 } = \frac{ -10 }{ 10 } + \frac{ 70 }{ 10 } i= -1+7i $$

Complex Division Calculator