Question

一道数学题,高手们快来帮忙!!!!!!!!!!!

已知2005X^3=2006Y^3=2007Z^3,XYZ>0,且(2005X^2+2006Y^2+2007Z^2)^(1/3)=2005^(1/3)+2006^(1/3)+2007^(1/3),求1/X+1/Y+1/Z的值.

Answer 1

令2005X^3=2006Y^3=2007Z^3=t
2005=t/x^3,2006=t/y^3,2007=t/z^3代入左边
2005^(1/3)=t^(1/3)/x,2006^(1/3)=t^(1/3)/y
2007^(1/3)=t^(1/3)/z代入右边
可得
(t/x+t/y+t/z)^(1/3)=t^(1/3)(1/x+1/y+1/z)
(1/x+1/y+1/z)^(1/3)=1/x+1/y+1/z
令1/x+1/y+1/z=w
w^3-w=0
w(w+1)(w-1)=0
w=1,w=-1,w=0

Answer 2

t/x+t/y+t/z)^(1/3)=t^(1/3)(1/x+1/y+1/z)
(1/x+1/y+1/z)^(1/3)=1/x+1/y+1/z
令1/x+1/y+1/z=w
w^3-w=0
w(w+1)(w-1)=0
w=1,w=-1,w=0