Question

How do you simplify $\frac{1}{2+3i}$?

Answer 1

$\frac{1}{2+3i}=\frac{2}{13}-\frac{3}{13}i$
Explanation:
$\frac{1}{2+3i}$ is reciprocal of complex number $2+3i$. To find reciprocal we multiply numerator and denominator each by complex conjugate of $2+3i$, which is $2-3i$. Hence,
$\frac{1}{2+3i}=\frac{1}{2+3i}\times \frac{2-3i}{2-3i}$
$=\frac{2-3i}{2^2-(3i{)}^2}$
$=\frac{2-3i}{4-9i^2}$
$=\frac{2-3i}{4+9}$
$=\frac{2}{13}-\frac{3}{13}i$