# Question #189e8

Oct 19, 2016

$\frac{3 i}{4 + 2 i} = \frac{3}{10} + \frac{3}{5} i$

#### Explanation:

The complex conjugate or conjugate of a complex number $a + b i$, denoted $\overline{a + b i}$, is given by

$\overline{a + b i} = a - b i$

A useful property of the complex conjugate is that for any complex number $z$, we have $z \overline{z} \in \mathbb{R}$, that is, the product of a complex number and is conjugate is real.

We use this property by multiplying the numerator and denominator of the expression by the conjugate of the denominator. This results in a real-valued denominator when we can then distribute (if necessary).

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

$= \frac{12 i - 6 {i}^{2}}{16 + 8 i - 8 i - 4 {i}^{2}}$

$= \frac{12 i - \left(- 6\right)}{16 - \left(- 4\right)}$

$= \frac{6 + 12 i}{20}$

$= \frac{3}{10} + \frac{3}{5} i$