# How do you simplify 7+4i div 2-3i?

Jan 12, 2016

$\frac{2}{13} + \frac{29}{13} i$

#### Explanation:

Basically, $\left(7 + 4 i\right) \div \left(2 - 3 i\right)$ is the same as the fraction $\frac{7 + 4 i}{2 - 3 i}$. As I prefer to work with fractions, I will stick with this formulation.

To simplify $\frac{7 + 4 i}{2 - 3 i}$, you need to find the complex conjugate of the denominator and extend the fraction with it:

Your denominator is $2 - 3 i$, so the complex conjugate is $2 + 3 i$.

You need to extend the fraction with $2 + 3 i$, i.e. multiply both the numerator and the denominator by it:

$\frac{7 + 4 i}{2 - 3 i} = \frac{\left(7 + 4 i\right) \cdot \left(2 + 3 i\right)}{\left(2 - 3 i\right) \cdot \left(2 + 3 i\right)} = \frac{14 + 8 i + 21 i + 12 {i}^{2}}{{2}^{2} - {\left(3 i\right)}^{2}} = \frac{14 + 29 i + 12 {i}^{2}}{4 - 9 {i}^{2}}$

... remember that ${i}^{2} = - 1$...

$= \frac{14 - 12 + 29 i}{4 + 9} = \frac{2 + 29 i}{13} = \frac{2}{13} + \frac{29}{13} i$