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#[{-1+sqrt(-3)}/2]^19+[{-1-sqrt(-3)}/2]^19=-1#

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Answer 1 · 2017-12-29T11:46:28

Taking #1,omega,omega^2 # as cube roots of unity.

So if #omega =(1+sqrt(-3))/2# then #omega^2 =(1-sqrt(-3))/2#

and #omega^3=omegaxxomega^2=1#

#1+omega+omega^2=0#

Hence
#LHS=[{-1+sqrt(-3)}/2]^19+[{-1-sqrt(-3)}/2]^1#

#=omega^19+[omega^2]^19#

#=omega^19+omega^38#

#=(omega^3)^6xxomega+(omega^3)^12xxomega^2#

#=(1)^6xxomega+(1)^12xxomega^2#

#=omega+omega^2=-1=RHS#