# What does (2i)/(4-5i) equal?

$\frac{2 i}{4 - 5 i} = - \frac{10}{41} + \frac{8}{41} i$
Multiply numerator and denominator by the conjugate $\left(4 + 5 i\right)$ of the denominator and simplify...
$\frac{2 i}{4 - 5 i} = \frac{2 i \left(4 + 5 i\right)}{\left(4 - 5 i\right) \left(4 + 5 i\right)} = \frac{- 10 + 8 i}{{4}^{2} + {5}^{2}} = \frac{- 10 + 8 i}{16 + 25} = \frac{- 10 + 8 i}{41} = - \frac{10}{41} + \frac{8}{41} i$