What is complex conjugate of # [ (3 + 8i)^4 ] / [ (1+i)^10 ]#?

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Answer 1 · 2016-01-10T21:35:31

#-165+721/32i#

First note that #(1+i)^2 = 1^2+2i+i^2 = 2i# and #(2i)^2 = -4#

So

#(1+i)^10 = (2i)^5 = (2i)^2(2i)^2(2i) = (-4)(-4)(2i) = 32i#

#(3+8i)^4 = (9+48i+64i^2)^2 = (-55+48i)^2#

#= (3025-5280i+2304i^2) = 721-5280i#

So:

#(3+8i)^4/(1+i)^10 = (721-5280i)/(32i) = -721/32i-5280/32 = -721/32i-165 = -165-721/32i#

So, reversing the sign of the coefficient of #i#, the Complex conjugate is:

#-165+721/32i#