令ax^3=by^3=cz^3=k,则:a=k/x^3、b=k/y^3、c=k//z^3. 注意到:1/x+1/y+1/z=1.
∴(ax^2+by^2+cz^2)^(1/3)=(k/x+k/y+k/z)^(1/3)
=k^(1/3)(1/x+1/y+1/z)^(1/3)=k^(1/3).
a^(1/3)+b^(1/3)+c^(1/3)=k^(1/3)/x+k^(1/3)/y+k^(1/3)/z
=k^(1/3)(1/x+1/y+1/z)=k^(1/3).
∴(ax^2+by^2+cz^2)^(1/3)=a^(1/3)+b^(1/3)+c^(1/3).