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Compute (1 + i)^{20}, where i^2 = -1.

 May 11, 2020
 #1
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(1 + i) ^ (20) = -1024.

 May 11, 2020
 #2
avatar+23245 
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(1 + i)20

 

Raising a complex number to a power is generally easiest if you write the complex number in  r·cis(theta)  form.

 

For the complex number  x + yi:

 

r  =  sqrt( x2 + y2 )  =  sqrt( 12 + 12 )  =  sqrt(2)

 

theta  =  tan-1( y/x )  =  tan-1( 1/1 )  =  tan-1( 1)  =  45o.                    (you can also use radians)

 

Therefore,  1 + i  =  sqrt(2) · cis( 45o )

 

Formula:  (x + yi)n  =  [ r·cis(theta) ]n  =  rn · cis( n · theta )

 

(1 + i)20  =  [ sqrt(2) · cis( 45o ) ]20  =  [sqrt(2)]20 · cis( 20 · 45o )  =  1024·cis( 900o )  =  1024·cis( 180o​ ) 

 

  =  1024·cos(180o) + 1024·i·sin(180o)  =  1024·(-1) + 1024·i·(0)  =  -1024 + 0i

 May 11, 2020

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