Question

设2000x三次方=2001y三次方=2002z三次方,xyz>0,且根号三次方2000x平方+2001y平方+2002

Answer 1

1.为了简化算式,设A=2000的立方根,B=2001的立方根,C=2002的立方根

A^3=2000,B^3=2001,C^3=2002

A^3x^3=B^3y^3=C^z^3

y=Ax/B

z=Ax/C

2000x^2+2002z^2+2001y^2

=A^3*x^2+C^3*(Ax/C)^2+B^3*(Ax/B}^2

=A^3*x^2+A^2*C*x^2+A^2*B*x^2

=A^2*X^2*(A+B+C)

(2000x^2+2002z^2+2001y^2)的立方根=A+B+C

2000x^2+2002z^2+2001y^2=(A+B+C)^3

2000x^2+2002z^2+2001y^2=A^2*X^2*(A+B+C)

A^2*x^2*(A+B+C)=(A+B+C)^3

A^2*x^2=(A+B+C)^2

Ax=A+B+C

1/x=A/(A+B+C)

y=Ax/B=(A+B+C)/B

1/y=B/(A+B+C)

z=Ax/C=(A+B+C)/C

1/z=C/(A+B+C

1/x+1/y+1/z=A/(A+B+C)+B/(A+B+C)+C/(A+B+C=1