设2000X^3=2001Y^3=2002Z^3,XYZ＞0 且2000X^3+2001Y^3+2002Z^3的和的立方根=2000的立方根+2001的立方根+2设2000X^3=2001Y^3=2002Z^3,XYZ＞0 且2000X^3+2001Y^3+2002Z^3的和的立方根=2000的立方根+2001的立方根+2002的立方根.求X

来源：学生作业帮助网编辑：作业帮时间：2024/11/14 23:50:59

设2000X^3=2001Y^3=2002Z^3,XYZ＞0 且2000X^3+2001Y^3+2002Z^3的和的立方根=2000的立方根+2001的立方根+2设2000X^3=2001Y^3=2002Z^3,XYZ＞0 且2000X^3+2001Y^3+2002Z^3的和的立方根=2000的立方根+2001的立方根+2002的立方根.求X
令2000x^3=2001y^3=2002z^3=N,题中方程立方得
2000x^2+2001y^2+2000z^2=（3次√2000+3次√2001+3次√2002）^3
两边同除以N
左边：(2000x^2+2001y^2+2000z^2)/N=2000x^2/N+2001y^2/N+2000z^2/N
=2000x^2/2000x^3+2001y^2/2001y^3+2000z^2/2002z^3
=1/x+1/y+1/z
右边：（3次√2000+3次√2001+3次√2002）^3/N
=[(3次√2000/3次√N)+(3次√2001/3次√N)+(3次√2002/3次√N)]^3
=[(3次√2000/3次√2000x^3)+(3次√2001/3次√2001y^3)+(3次√2002/3次√2002z^3)]
=(1/x+1/y+1/z)^3
即是 1/x+1/y+1/z=(1/x+1/y+1/z)^3
化简 (1/x+1/y+1/z)^2=1
因为 2000x^3=2001y^3=2002z^3且xyz大于0
所以 x>0,y>0,z>0
所以 1/x+1/y+1/z=1

设2000x3=2001y3=2002z3=k 则2000x2=k/x 2001y2=k/y 2002z2=k/z
2000的立方根=k的立方根/x 2001的立方根=k的立方根/y 2002的立方根=k的立方根/z
由已知可得：（k/x+k/y+k/z)的立方根=k的立方根/x + k的立方根/y...

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设2000x3=2001y3=2002z3=k 则2000x2=k/x 2001y2=k/y 2002z2=k/z
2000的立方根=k的立方根/x 2001的立方根=k的立方根/y 2002的立方根=k的立方根/z
由已知可得：（k/x+k/y+k/z)的立方根=k的立方根/x + k的立方根/y +k的立方根/z
所以就可以得到(1/x+1/y+1/z)的立方根=1/x+1/y+1/z
又因为xyz>0,所以只能是x>0 y>0 z>0
所以1/x+1/y+1/z=1

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