Find the product of the complex number and its conjugate. -6 - 6i

Answer 1

The conjugate of -6 - 6i is -6 + 6i. If you multiply these together the result is a difference of squares.

For example:

(x - 6)(x + 6) = x² - 36

For the given problem:

(-6)² - (6i)² = 36 - 36i²; since i²=-1 this is: 36 + 36 = 72.

Whenever you multiply complex conjugates the result is a complex number with an imaginary part equal to zero.

This fact is what accounts for the technique used to divide complex numbers.

To divide (3 + 5i) by (2 - 3i) (which can be written as (3 + 5i) / (2 -3i) multiply numerator and denominator of that fraction by the denominator's conjugate:

(3 + 5i)(2 + 3i) / (2 - 3i)(2 + 3i) = [6+ 9i + 10i +15i²] / [4 - 9i²] = (-9 +19i) / 13 which can also be written as:

-9/13 + (19/13)i