# How do you divide (2i)/(4-5i)?

Dec 3, 2015

$- \frac{10}{41} + \frac{8}{41} i$

#### Explanation:

Multiply by the complex conjugate.

$\frac{2 i}{4 - 5 i} \left(\frac{4 + 5 i}{4 + 5 i}\right) = \frac{8 i + 10 {i}^{2}}{16 + 20 i - 20 i - 25 {i}^{2}} = \frac{8 i + 10 {i}^{2}}{16 - 25 {i}^{2}}$

Recall that $i = \sqrt{- 1}$, so ${i}^{2} = - 1$.

$\frac{8 i + 10 \left(- 1\right)}{16 - 25 \left(- 1\right)} = \frac{8 i - 10}{16 + 25} = \frac{- 10 + 8 i}{41}$

In the form of a complex number, this would be written as:

$- \frac{10}{41} + \frac{8}{41} i$