# How do you simplify i/(7+4i) ?

Mar 6, 2016

$\frac{4}{65} + \frac{7}{65} i$

#### Explanation:

In order to remove the complex number from the denominator, we need to multiply the fraction by the complex conjugate.

$= \frac{i}{7 + 4 i} \left(\frac{7 - 4 i}{7 - 4 i}\right)$

Distribute.

$= \frac{7 i - 4 {i}^{2}}{49 + 28 i - 28 i - 16 {i}^{2}}$

We can simplify this knowing that since $i = \sqrt{- 1} \implies \underline{{i}^{2} = - 1}$.

$= \frac{7 i - 4 \left(- 1\right)}{49 - 16 \left(- 1\right)}$

$= \frac{4 + 7 i}{49 + 16}$

$= \frac{4 + 7 i}{65}$

$= \frac{4}{65} + \frac{7}{65} i$