Question

How do you divide `4/(-3-6i)`?

Answer 1

`-4/15+8/15i`

Explanation:

To divide this fraction we require the denominator to be `color(blue)"real"`

To obtain this multiply (-3 - 6i) by it's `color(blue)"conjugate"`

The conjugate of (-3 - 6i) is (-3 + 6i). Note that the real part remains unchanged while the `color(red)"sign"` of the imaginary part is reversed.

`rArr(-3-6i)(-3+6i)=9-18i+18i-36i^2=45" a real number"`

`color(orange)"Reminder " color(red)(|bar(ul(color(white)(a/a)color(black)(i^2=(sqrt(-1))^2=-1)color(white)(a/a)|)))`

Since this is a fraction, we also have to multiply the numerator by (-3 + 6i)

`4/(-3-6i)xx(-3+6i)/(-3+6i)=(4(-3+6i))/((-3-6i)(-3+6i))`

`=(-12+24i)/45=-12/45+24/45i=-4/15+8/15i`