Question

非零复数a,b满足a^2+ab+b^2=0求(a/(a+b))^2003+(b/(a+b))^2003=

Answer 1

由a^2+ab+b^2=0,a/(a+b)=-b/a,b/(a+b)=-a/b

a^3-b^3=0

(a/b)^3=(b/a)^3=1

(a/(a+b))^2003+(b/(a+b))^2003=-(b/a)^2003-(a/b)^2003=-(b/a)^2-(a/b)^2

a^2+ab+b^2=0

a^4+2a^2*b^2+b^4=(-ab)^2,a^4+b^4=-a^2*b^2

(a/(a+b))^2003+(b/(a+b))^2003=-(a^4+b^4)/(a^2*b^2)=1