How can evaluate?(1+i)^2008=?

(1+i)^2008.

To simplify,
first we will rewrite the exponent.

==> ( 1+ i)^2008
= (1+ i)^(2*1004)

Now we know from exponent properties that
x^ab= (x^a)^b

==> (1+i)^(2*1004) =[
(1+i)^2]^1004

Now we will expand the
brackets.

==> (1+i)^2]^1004 = (1 + 2i+ i^2)
^1004

But we know that i^1 =
-1

==> (1+2i+i^2)^1004 =
(2i)^1004

==> (2i)^(2*502) =[ (2i)^2]^502 =
(4i^2)^502

==> (4*-1)^502 = =
4^502

==> (1+i)^2008 =
2^(1004)