# How do you simplify 1/(2+3i)?

Jun 16, 2018

$\frac{1}{2 + 3 i} = \frac{2}{13} - \frac{3}{13} i$

#### Explanation:

$\frac{1}{2 + 3 i}$ is reciprocal of complex number $2 + 3 i$. To find reciprocal we multiply numerator and denominator each by complex conjugate of $2 + 3 i$, which is $2 - 3 i$. Hence,

1/(2+3i)=1/(2+3i)×(2-3i)/(2-3i)

= $\frac{2 - 3 i}{{2}^{2} - {\left(3 i\right)}^{2}}$

= $\frac{2 - 3 i}{4 - 9 {i}^{2}}$

= $\frac{2 - 3 i}{4 + 9}$

= $\frac{2}{13} - \frac{3}{13} i$

Jun 16, 2018

$\frac{2}{13} - \frac{3}{13} i$

#### Explanation:

$\text{multiply the numerator/denominator by the complex}$
$\text{conjugate of the denominator}$

$\text{the conjugate of "2+3i" is } 2 \textcolor{red}{-} 3 i$

$\left(2 + 3 i\right) \left(2 - 3 i\right) = {2}^{2} - 9 {i}^{2} = 4 + 9 = 13$

$= \frac{2 - 3 i}{13} = \frac{2}{13} - \frac{3}{13} i \leftarrow \textcolor{b l u e}{\text{in standard form}}$