When dividing complex numbers (4 + 5i) / (2 - 2i), which operation should be performed?

To divide complex numbers such as (4 + 5i) by (2 - 2i), the appropriate operation involves multiplying by the conjugate of the denominator. The conjugate of a complex number is obtained by changing the sign of its imaginary part. For the denominator (2 - 2i), the conjugate is (2 + 2i).

When you multiply both the numerator and the denominator by the conjugate of the denominator, it allows you to eliminate the imaginary part from the denominator. This results in a real number in the denominator, making the division straightforward. So, for (4 + 5i)/(2 - 2i), you would perform the multiplication as follows:

(4 + 5i) * (2 + 2i) and (2 - 2i) * (2 + 2i).

This process ultimately leads to a simpler form of the quotient expressed as a complex number, with both a real and an imaginary component. Hence, multiplying by the conjugate of the denominator is the necessary operation to perform when dividing complex numbers in this manner.