Question

设a的x次方=b的y次方=（ab）的z次方,且xyz不等于0,a和b均为不等于1的正数,证明z=x+y分之xy

Answer 1

a^x=(ab)^z=a^z*b^z

a^(x-z)=b^z

b=a^[(x-z)/z] (1)

b^y=(ab)^z=a^z*b^z

b^(y-z)=a^z

b=a^[z/(y-z)] (2)

(1)=(2)

所以a^[(x-z)/z]=a^[z/(y-z)]

即(x-z)/z=z/(y-z)

z²=(x-z)(y-z)=xy-xz-yz+z²

z(x+y)=xy

故z=xy/(x+y)

得证