非零复数a,b满足a^2+ab+b^2=0,则（a/(a+b))^1999+（b/(a+b))^1999的值是?

非零复数a,b满足a^2+ab+b^2=0,则（a/(a+b))^1999+（b/(a+b))^1999的值是?
非零复数a,b满足a^2+ab+b^2=0,则（a/(a+b))^1999+（b/(a+b))^1999的值是?

非零复数a,b满足a^2+ab+b^2=0,则（a/(a+b))^1999+（b/(a+b))^1999的值是?
复数a,b非零,且
a^2 + ab + b^2 = 0,
所以,
(a+b)^2 = ab
a/(a+b) = (a+b)/b = [b/(a+b)]^(-1) ...（1）
又,
a/(a+b) + b/(a+b) = 1 ...（2）
令 u = a/(a+b),则由（1）和（2）解得,
a/(a+b) = u = (1 + i(3)^(1/2))/2
= exp[iPI/3],
b/(a+b) = (1 - i(3)^(1/2))/2
= exp[-iPI/3]
或者,
a/(a+b) = u = (1 - i(3)^(1/2))/2
= exp[-iPI/3],
b/(a+b) = (1 + i(3)^(1/2))/2
= exp[iPI/3].
所以,总有,
（a/(a+b))^1999 +（b/(a+b))^1999
= {exp[iPI/3]}^1999 + {exp[-iPI/3]}^1999
= exp[iPI(1999/3)] + exp[-iPI(1999/3)]
= exp[iPI/3] + exp[-iPI/3]
= (1 + i(3)^(1/2))/2 + (1 - i(3)^(1/2))/2
= 1