+0  
 
+4
121
1
avatar+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. 

 Dec 20, 2019
 #1
avatar+109563 
+3

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

 

 

 

cool cool cool

 Dec 20, 2019

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