Question

How do you write the following quotient in standard form `4/(4-5i)`?

Answer 1

`16/41+20/41i`

Explanation:

To perform the division, we require to multiply the numerator/denominator by the `color(blue)"complex conjugate"` of the denominator. This ensures that we have a rational value on the denominator.

The conjugate of `4-5i" is "4color(red)(+)5i`

`rArr4/(4-5i)=(4(4+5i))/((4-5i)(4+5i))=(16+20i)/41`

`color(blue)"Evaluating denominator"`

`rarr[(4-5i)(4+5i)=16-20i+20i-25i^2=41]larr`

`color(orange)"Reminder "i^2=(sqrt(-1))^2=-1`

`rArr4/(4-5i)=(16+20i)/41=16/41+20/41i`