设2008x³=2009y³=2010z³,xyz>0,且(2008x²+2009y²+2010z²)的立方根=2008

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  • 2008x³=2009y³=2010z³ .条件①

    2008x²+2009y²+2010z²)的立方根=2008的立方根+2009的立方根+2010的立方根 .条件②

    令2008x^3=2009y^3=2010^3=M

    可以知道:2008x^2=M/x 2009y^2=M/y 2010z^2=M/z

    又(2008x²+2009y²+2010z²)=M(1/x+1/y+1/z)

    故:〔M(1/x+1/y+1/z)〕^(1/3)=M^(1/3)*(1/x+1/y+1/z)^(1/3)

    同时2008^(1/3)=M^(1/3)/x 2009^(1/3)=M^(1/3)/y

    2010^(1/3)=M^(1/3)/z

    综合条件二可知:

    M^(1/3)*(1/x+1/y+1/z)^(1/3)=M^(1/3)/x+M^(1/3)/y+M^(1/3)/z

    消去M^(1/3)可得:

    (1/x+1/y+1/z)^(1/3)=1/x+1/y+1/z

    这样我们就可以知道结果是

    1/x+1/y+1/z=1