Ta có: \(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}=\dfrac{1}{x+y+z}\)

\(\Leftrightarrow\dfrac{xy+yz+xz}{xyz}=\dfrac{1}{x+y+z}\)

\(\Leftrightarrow\left(xy+yz+xz\right)\left(x+y+z\right)=xyz\)

\(\Leftrightarrow x^2y+xy^2+y^2z+yz^2+x^2z+xz^2+3xyz-xyz=0\)

\(\Leftrightarrow x^2y+xy^2+y^2z+yz^2+x^2z+xz^2+2xyz=0\)

\(\Leftrightarrow x^2y+xy^2+x^2z+xyz+y^2z+yz^2+xz^2+xyz=0\)

\(\Leftrightarrow x\left(xy+y^2+xz+yz\right)+z\left(y^2+yz+xz+xy\right)=0\)

\(\Leftrightarrow x\left[y\left(x+y\right)+z\left(x+y\right)\right]+z\left[y\left(y+z\right)+x\left(y+z\right)\right]=0\)

\(\Leftrightarrow x\left(x+y\right)\left(y+z\right)+z\left(y+z\right)\left(x+y\right)=0\)

\(\Leftrightarrow\left(x+y\right)\left(y+z\right)\left(z+x\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}x+y=0\\y+z=0\\z+x=0\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=-y\\y=-z\\z=-x\end{matrix}\right.\)

* x = -y

\(\dfrac{1}{x^{2007}}+\dfrac{1}{y^{2007}}+\dfrac{1}{z^{2007}}=\dfrac{1}{x^{2007}}-\dfrac{1}{x^{2007}}+\dfrac{1}{z^{2007}}=\dfrac{1}{z^{2007}}\)(*)

\(\dfrac{1}{x^{2007}+y^{2007}+z^{2007}}=\dfrac{1}{x^{2007}-x^{2007}+z^{2007}}=\dfrac{1}{z^{2007}}\)(*)

Từ (*) và (**) \(\Rightarrow\) đpcm

Tương tự xét y = -z và z = -x

Vậy nếu x, y, z khác 0 và x + y +z khác 0 thì \(\dfrac{1}{x^{2007}}+\dfrac{1}{y^{2007}}+\dfrac{1}{z^{2007}}=\dfrac{1}{x^{2007}+y^{2007}+z^{2007}}\).