How do you use DeMoivre's theorem to simplify #(1+i)^4#?

1 Answer
George C.
Sep 11, 2016

#(1+i)^4 = -4#

Explanation:

Note that

#1+i = sqrt(2)(cos (pi/4) + i sin (pi/4))#

De Moivre tells us that:

#(cos theta + i sin theta)^n = cos n theta + i sin n theta#

So we find:

#(1+i)^4 = (sqrt(2)(cos (pi/4) + i sin (pi/4)))^4#

#color(white)((1+i)^4) = (sqrt(2))^4(cos (pi/4) + i sin (pi/4))^4#

#color(white)((1+i)^4) = 4(cos pi + i sin pi)#

#color(white)((1+i)^4) = 4((-1) + i (0))#

#color(white)((1+i)^4) = -4#

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