已知ax^3=by^3=cz^3,三次根号(ax^2+by^2+cz^2)=三次根号a+三次根号b+三次根号c,求(1/x+1/y+1/z)的值

已知ax^3=by^3=cz^3,三次根号(ax^2+by^2+cz^2)=三次根号a+三次根号b+三次根号c,求(1/x+1/y+1/z)的值
已知ax^3=by^3=cz^3,三次根号(ax^2+by^2+cz^2)=三次根号a+三次根号b+三次根号c,求(1/x+1/y+1/z)的值

已知ax^3=by^3=cz^3,三次根号(ax^2+by^2+cz^2)=三次根号a+三次根号b+三次根号c,求(1/x+1/y+1/z)的值
(ax^2+by^2+cz^2)^(1/3)=a^(1/3)+b^(1/3)+c^(1/3)
(ax^3(1/x+1/y+1/z))^(1/3)=a^(1/3)+b^(1/3)+c^(1/3)
x=[a^(1/3)+b^(1/3)+c^(1/3)]/(a(1/x+1/y+1/z)^(1/3)
1/x=(a(1/x+1/y+1/z)^(1/3)/[a^(1/3)+b^(1/3)+c^(1/3)] (1)
同理：
1/y=(b(1/x+1/y+1/z)^(1/3)/[a^(1/3)+b^(1/3)+c^(1/3)] (2)
1/z=(c(1/x+1/y+1/z)^(1/3)/[a^(1/3)+b^(1/3)+c^(1/3)] (3)
(1)+(2)+(3)得
1/x+1/y+1/z=(1/x+y/1+1/z)^(1/3)
(1/x+1/y+1/z)((1/x+1/y+1/z)^2-1)=0
所以1/x+1/y+1/z=0 或1/x+1/y+1/z=1,或1/x+1/y+1/z=-1