Question

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

Answer 1

`3/5+4/5i`

Explanation:

When dividing complex numbers , to simplify we rationalise the denominator. This means changing it into a rational number instead of a complex number.

This is achieved by multiplying the numerator and denominator by the `color(blue)" conjugate of the denominator"`

Given a complex number a + bi , conjugate is a - bi

Note that a and b remain unchanged whist the sign changes.

In general conjugate of a ± bi is a ∓ bi

here the conjugate of 4 - 2i is 4 + 2i

`rArr(4+2i)/(4-2i)xx(4+2i)/(4+2i)=((4+2i)(4+2i))/((4-2i)(4+2i))`

expanding numerator and denominator.

`=(16+16i+4i^2)/(16-4i^2)`

Using `( i^2=-1)` this now becomes

`(16+16i-4)/(16+4)=(12+16i)/20=12/20+16/20i=3/5+4/5i`