Question

设非零复数x,y满足x^2+xy+y^2=0,求代数式（x/x+y）^1990+(y/x+y)^1990的值

Answer 1

x^2=-y(x+y)

y^2=-x(x+y)

所以x/(x+y)=-y/x

y/(x+y)=-x/y

而(x/y)^2+x/y+1=0

所以(x/y)^3=1

同样(y/x)^3=1`

所以[x/(x+y)]^1990+[y/(x+y)]^1990

=(y/x)^1990+(x/y)^1990

=(y/x)^2+(x/y)^2

=-1-y/x-1-x/y

=-2-(x^2+y^2)/xy

=-1