+41 Simplify:
\({\frac{(1+i)^{2011}}{(1-i)^{2009}}}\)
I've tried splitting up the exponents into smaller groups; for example I split 2011 into 2 * 1005 + 1.
But, I'm stuck after this point.
+128474 Not as hard as it seems
(1 + i)^2011
___________
( 1 - i)^2009
Note that ( 1 + i)^2 = 2i
And ( 1 - i)^2 = -2i
So we have
[ (1 + i)^2 ] ^1005 * ( 1 + i)
_______________________ =
[ ( 1 - i)^2]^1004 * ( 1 - i)
[2i ] ^1005 * (1 + i)
________________ =
[ -2i]^ 1004 * ( 1 - i)
[ 2i] * [ 2i]^1004 * ( 1 + i)
_____________________
[ 2i]^1004 * ( 1 - i)
2i ( 1 + i)
________
( 1 - i)
2i ( 1 + i) ( 1 + i)
_____________
(1 - i) ( 1 + i)
2i ( 2i)
________
1 - i^2
4i^2
_______
1 - (-1)
-4
____
2
-2
