Question

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

Answer 1

`2/5+1/5i`

Explanation:

To simplify this fraction, we must multiply the numerator/denominator by the `color(blue)"complex conjugate"` of the denominator.
This ensures that the denominator is a real number.

Given a complex number `z=x±yi` then the complex conjugate is.

`color(red)(bar(ul(|color(white)(2/2)color(black)(barz=x∓yi)color(white)(2/2)|)))`

Note that the real part remains unchanged while the `color(red)"sign"` of the imaginary part is reversed.

Hence the conjugate of `6-3i" is " 6+3i`

and `(6-3i)(6+3i)=36+9=45larr" a real number"`

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

Multiply numerator/denominator by 6 + 3i

`(3(6+3i))/((6-3i)(6+3i))=(18+9i)/45=18/45+9/45i=2/5+1/5i`