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

Mar 6, 2018

Multiply both the top and bottom by the conjugate of the denominator, $\left(2 - i\right)$, and simplify to get $\frac{8}{5} - \frac{1}{5} i$

#### Explanation:

Starting with $\frac{3 + 2 i}{2 + i}$, we can get $i$ out of the denominator by multiplying both the numerator and denominator by the "conjugate" of the denominator, which is just the denominator with the sign switched in the middle:
$\frac{3 + 2 i}{2 + i} \cdot \frac{2 - i}{2 - i}$
Multiply and simplify (remember that $i = \sqrt{- 1}$ so ${i}^{2} = - 1$) to get

$\frac{6 - 3 i + 4 i - 2 {i}^{2}}{4 - {i}^{2}} = \frac{6 + i - 2 \left(- 1\right)}{4 - \left(- 1\right)} = \frac{6 + i + 2}{5} = \frac{8 + i}{5}$

Finally, your teacher may or may not care about this, but "standard form" for a complex number is $a \pm b i$, where $a$ is the Real number part. Put your answer in this format by breaking up the numerator:
$\frac{8 + i}{5} = \frac{8}{5} - \frac{1}{5} i$