# How do you simplify (8 i)/(2-i)?

May 20, 2016

$\frac{8 i}{2 - i} = - \frac{8}{5} + \frac{16}{5} i$

#### Explanation:

Given a complex number $a + b i$, the complex conjugate of that number is $a - b i$. A useful property is that the product of a complex number and its conjugate is a real number.

In this case, we will use that property, along with the fact that the conjugate of $2 - i$ is $2 + i$, to remove the complex number from the denominator.

$\frac{8 i}{2 - i} = \frac{8 i \left(2 + i\right)}{\left(2 - i\right) \left(2 + i\right)}$

$= \frac{- 8 + 16 i}{5}$
$= - \frac{8}{5} + \frac{16}{5} i$