# How do you divide (2 + i) / ( 5- i ) ?

Mar 15, 2018

$\frac{9 + 7 i}{26}$

#### Explanation:

Dividing complex numbers is similar to rationalizing the denominator of a surd.

$\frac{2 + i}{5 - i} = \frac{\left(2 + i\right) \textcolor{g r e e n}{\left(5 + i\right)}}{\left(5 - i\right) \textcolor{g r e e n}{\left(5 + i\right)}}$

=(10+5i+2i+i^2)/(25+5i-5i-i^2

Recalling that ${i}^{2} = - 1$

$= \frac{9 + 7 i}{26}$

Mar 15, 2018

$= \frac{9}{26} + \frac{7 i}{26}$

#### Explanation:

If you multiply a given fraction by 1 you do not change it.
Also, if a fraction has the same numerator and denominator it equals 1.
Now lets apply the following trick:

$\frac{2 + i}{5 - i} = \frac{2 + i}{5 - i} \cdot \textcolor{red}{\frac{5 + i}{5 + i}}$

$= \frac{10 + 5 i + 2 i + {i}^{2}}{{5}^{2} - {i}^{2}}$

$= \frac{10 + 7 i - 1}{25 + 1}$

$= \frac{9 + 7 i}{26}$

$= \frac{9}{26} + \frac{7 i}{26}$