What is (1 – 2i) ^6?

Given the complex number (
1-2i)^6

We need to
simplify.

==> We will rewrite the
power.

==> ( 1-
2i)^(2*3)

But we know that (x)^ab=
(x^a)^b

==> (1-2i)^(2*3) =
(1-2i)^2]^3

Now we will open the
brackets.

==> (1 - 4i +
4i^2)^3

But i^2 =-1

==>
( 1- 4i -4)^3

==> ( -3 -4i)^3 = (-3-4i)^2 *
(-3-4i)

                         = ( 9 +24i + 16i^2) *
(-3-4i)

                       = ( 9+24i -16) *
(-3-4i)

                       = (
-7+24i)*(-3-4i)

                       = ( 21 -72i +28i -
96i^2)

                       = ( 21+96 -
44i)

                      = ( 117
-44i)

==> ( 1-2i)^6 = 117 -
44*i