已知X,Y是非零复数,X^2+XY+Y^2=0,[X/(X+Y）]^2005+[Y/(X+Y)]^2005=?_数学解答

已知X,Y是非零复数,X^2+XY+Y^2=0,[X/(X+Y）]^2005+[Y/(X+Y)]^2005=?

人气：139 ℃　时间：2020-02-05 20:29:18

解答

x²+y²=-xy
(x+y)²=xy
(x-y)(x²+xy+y²)=0
x³-y³=0
x³=y³
(x/y)³=1
x^2005=(x³)^668*x=(y³)^668*x
所以原式=(x^2005+y^2005)/(x+y)^2005
=(x+y)*(x³)^668/(x+y)^2005
=x^2004/[(x+y)²]^1002
=x^2004/(xy)^1002
=(x/y)^1002
=[(x/y)³]^334
=1^334
=1