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

Jan 31, 2016

To simplify a complex fraction, multiply top and bottom (denominator and numerator) by the complex conjugate of the bottom. This expression simplifies to $\frac{2 - 5 i}{29}$.

#### Explanation:

The 'complex conjugate' of a complex number $\left(a + b i\right)$ is $\left(a - b i\right)$, and multiplying a complex number by its complex conjugate will yield a real number.

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

Note that

$\frac{2 - 5 i}{2 - 5 i} = 1$ so we are not changing the original number by simplifying it.

$\frac{1}{2 + 5 i} \cdot \frac{2 - 5 i}{2 - 5 i} = \frac{2 - 5 i}{4 + 10 i - 10 i - 25 {i}^{2}} = \frac{2 - 5 i}{4 - 25 {i}^{2}}$

Note that ${i}^{2} = - 1$ because $i = \sqrt{- 1}$

So the simplified version of this expression is

$\frac{2 - 5 i}{4 - \left(- 25\right)} = \frac{2 - 5 i}{29}$