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}{ 292+170i }\, $ is $ \color{blue}{ 292-170i } $.

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

$$\begin{aligned} \frac{ 142129 }{ 292+170i } &= \frac{ 142129 }{ 292+170i } \cdot \frac{ \color{blue}{ 292-170i } }{ \color{blue}{ 292-170i } } = \\[1 em] &= \frac{ 142129 \cdot \left( 292-170i \right) }{ \left( 292+170i \right) \cdot \left( 292-170i \right) } \end{aligned} $$

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

$$ \begin{aligned} 142129 \cdot \left( 292-170i \right) &= 142129 \cdot 292 + 142129 \cdot \left(-170 \,i \right) = \\[1 em] &= 41501668 -24161930 \, i = \\[1 em] &= 41501668-24161930i\end{aligned} $$ $$ \begin{aligned} \left( 292+170i \right) \cdot \left( 292-170i \right) &= 292 \cdot 292 + 292 \cdot \left(-170 \,i \right) + \left( 170 \,i \right) \cdot \left(292 \right) + \left( 170 \,i \right) \cdot \left(-170 \,i \right) = \\[1 em] &= 85264 -49640 \, i + 49640 \, i -28900 \color{blue}{(-1)} = \\[1 em] &= 114164\end{aligned} $$

Step 4: Separate real and imaginary parts:

$$ \frac{ 142129 }{ 292+170i } = \frac{ 41501668-24161930i }{ 114164 } = \frac{ 41501668 }{ 114164 } + \frac{ -24161930 }{ 114164 } i= \frac{ 10375417 }{ 28541 }-\frac{ 12080965 }{ 57082 }i $$

Complex number calculator