Question

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

Answer 1

`(9+7i)/26`

Explanation:

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

`(2+i)/(5-i)=((2+i)color(green)((5+i)))/((5-i)color(green)((5+i)))`

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

Recalling that `i^2=-1`

`=(9+7i)/26`

Answer 2

`=9/26+(7i)/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:

`(2+i)/(5-i)=(2+i)/(5-i)*color(red)((5+i)/(5+i))`

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

`=(10+7i-1)/(25+1)`

`=(9+7i)/26`

`=9/26+(7i)/26`