非零复数a,b满足a^2+ab+b^2=0求(a/(a+b))^2003+(b/(a+b))^2003=
问题描述：
非零复数a,b满足a^2+ab+b^2=0求(a/(a+b))^2003+(b/(a+b))^2003=
答
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