# How do you simplify (1+2i)/(2-3i)?

Jan 12, 2016

$\frac{1 + 2 i}{2 - 3 i} = \frac{- 4 + 7 i}{13} = - \frac{4}{13} + \frac{7}{13} i$

#### Explanation:

1. Find the complex coniugate of denominator

denominator: $z = 2 - 3 i$

denominator complex coniugate: $\overline{z} = 2 \textcolor{red}{+} 3 i$

1. Multiply both numerator and denominator for the complex coniugate

$\frac{1 + 2 i}{2 - 3 i} \cdot \frac{2 + 3 i}{2 + 3 i} = \frac{2 + 3 i + 4 i + 6 {i}^{2}}{{2}^{2} - {\left(3 i\right)}^{2}} =$

Remembering that: ${i}^{2} = - 1$

$= \frac{2 + 6 \cdot \left(- 1\right) + 7 i}{4 - 9 {i}^{2}} = \frac{2 - 6 + 7 i}{4 - 9 \cdot \left(- 1\right)} = \frac{- 4 + 7 i}{4 + 9} =$
$= \frac{- 4 + 7 i}{13} = - \frac{4}{13} + \frac{7}{13} i$