x√(yz)+y√(xz)=39-xy => xy+x√(yz)+y√(xz)=√(xy)*[√(xy)+√(xz)+√(yz)]=39(1)
y√(xz)+z√(xy)=52-yz => yz+y√(xz)+z√(xy)=√(yz)*[√(yz)+√(xy)+√(xz)]=52(2)
z√(xy)+x√(yz)=78-xz => xz+z√(xy)+x√(yz)=√(xz)*[√(xz)+√(yz)+√(xy)]=78 (3)
(1)/(2), => √(x/z)=39/52=3/4(4)
(2)/(3), => √(y/x)=52/78=2/3(5)
(1)/(3), => √(y/z)=39/78=1/2(6)
(4)*(6) => √(xy)=3/8*z,(7)
(4)*z => √(xz)=3/4*z, (8)
(6)*z => √(yz)=1/2*z(9)
将(7),(8),(9)代入(1),得3/8*z[3/8*z+3/4*z+1/2*z]=39
可解得 z=8,再由(8), (9) 解得 x=9/2, y=2
∴ 方程的解为 x=9/2, y=2, z=8