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Simplify (1+i)^2016-(1-i)^2016.

 Oct 7, 2018
 #1
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(1+i)^2016 =((1+i)^16)^126 =256^126

(1 - i)^2016=((1-i)^16)^126 =256^126

256^126 - 256^126 =0

 Oct 7, 2018
 #2
avatar+101872 
+1

( 1 + i)^2  =  1 + 2i  + i^2  =  1  + 2i  - 1  = 2i

 

(1 - i)^2  =  1 - 2i  + i^2  =  1  - 2i  - 1  =  -2i

 

So

 

( 1 + i)^2016  - (1 - i)^2016  =

 

[ (1 + i)^2 ] ^1008  -  [ (1 - i)^2 ]^1008  =

 

(2i)^1008  - (-2i)^1008  -

 

(2i)^1008  -  (-2)^1008 * (i)^1008  =           [ (-2)^even power  = 2^even power]

 

(2i)^1008  -  (2)^1008 * i^1008  =

 

(2i)^1008  -  (2i)^1008  =

 

0

 

 

cool cool cool

 Oct 8, 2018

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