Question

How do you divide `(6-i) / (7-2i)`?

Answer 1

`44/53 + 5/53 i`

Explanation:

To divide this fraction we require to rationalise the denominator.

We do this by multiplying numerator/denominator by the`color(blue)" complex conjugate " " of the denominator "`

If `color(blue)" a ± bi " " is a complex number then "`

`color(red)" a ∓ bi " " is it's conjugate "`

Note that the 'real part' remains unchanged , while the sign of the 'imaginary part' changes.

Also (a+ bi)(a - bi) = `a^2 - b^2 " a real number "`

and `i^2 = (sqrt(-1))^2 = -1`

Now the conjugate of 7 - 2i is 7 + 2i

multiplying numerator / denominator by (7 + 2i)

`rArr ((6 - i)(7 + 2i))/((7 - 2i)(7 + 2i)) = (42 + 5i +2)/(49 + 4)= 44/53 + 5/53 i`