Question

How do you divide `1/(-8-5i)`?

Answer 1

`-8/89+5/89i`

Explanation:

To divide this fraction we require to make the denominator real.

This can be achieved by multiplying (-8 - 5i) by it's `color(blue)"conjugate"`

If `z=x±yi` then the conjugate is.

`color(orange)"Reminder " color(red)(bar(ul(|color(white)(a/a)color(black)(barz=x∓yi)color(white)(a/a)|)))`
Note that x, remains unchanged while the sign of the imaginary part is reversed.

`rArr(-8+5i)" is the conjugate of" -8-5i`

and `(-8-5i)(-8+5i)=64-25i^2=64+25=89`

That is we have a real value on the denominator.

Since this is a fraction, we must of course multiply both numerator and denominator by the conjugate.

`rArr(-8+5i)/((-8-5i)(-8+5i))=(-8+5i)/89=-8/89+5/89i`