# How do you simplify (4+5i)/(2-3i)?

Feb 13, 2016

$\frac{4 + 5 i}{2 - 3 i} = - \frac{7}{13} + \frac{22}{13} i$

#### Explanation:

Given a complex number $z = a + b i$, the complex conjugate of $z$, denoted $\overline{z}$, is $\overline{z} = a - b i$. For any complex number, $z \overline{z}$ is a real number. Thus, we can eliminate the complex number from the denominator by multiplying the numerator and the denominator by the conjugate of the denominator.

$\frac{4 + 5 i}{2 - 3 i} = \frac{\left(4 + 5 i\right) \left(2 + 3 i\right)}{\left(2 - 3 i\right) \left(2 + 3 i\right)}$

$= \frac{- 7 + 22 i}{13}$

$= - \frac{7}{13} + \frac{22}{13} i$