Question

How do you simplify `(2-4i)/(4-3i)`?

Answer 1

`4/5 - 2/5 i`

Explanation:

To simplify , require the denominator to be real. To achieve this multiply the numerator and denominator by the 'complex conjugate' of the denominator.

If (a + bi ) is a complex number then (a - bi ) is the conjugate.

Note that : (a + bi )(a - bi ) = `a^2 + b^2color(black)(" which is real ")`

here the conjugate of (4 - 3i ) is (4 + 3i )

hence : `((2 - 4i)(4 + 3i))/((4 -3i)(4 + 3i ))`

distribute using FOIL

`= (8+6i-16i -12i^2 )/(16 + 12i - 12i - 9i^2 )`

[noting that: `i^2 =(sqrt-1)^2 = -1`]

`= (8 - 10i + 12)/(16 + 9 ) = (20 - 10i )/25 = 20/25 - 10/25 i = 4/5 - 2/5 i`