# How do you simplify 1/(2+5i )?

Nov 7, 2015

$\frac{2 - 5 i}{29}$

#### Explanation:

Multiply by a special form of $1$. To figure out what that special form of $1$ is, we take the complex conjugate of the bottom divided by itself.

That's just a fancy way of saying switch the sign in front of the imaginary term and do this:

$\frac{2 - 5 i}{2 - 5 i}$

now we multiply our original expression by that:

$\frac{1}{2 + 5 i} \cdot \frac{2 - 5 i}{2 - 5 i}$

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

$\frac{2 - 5 i}{4 - 25 {i}^{2}}$

$\frac{2 - 5 i}{4 + 25}$

$\frac{2 - 5 i}{29}$