Ta có : \(\left(x+y+z\right)^2=x^2+y^2+z^2\)
\(\Rightarrow x^2+y^2+z^2+2\left(xy+yz+zy\right)=x^2+y^2+z^2\)
\(\Rightarrow2\left(xy+yz+zx\right)=0\)
\(\Rightarrow xy+yz+zx=0\)
\(\Rightarrow\frac{xy}{xyz}+\frac{yz}{xyz}+\frac{zx}{xyz}=0\)( Chia 2 vế cho xyz )
\(\Rightarrow\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=0\)
\(\Rightarrow\frac{1}{x}+\frac{1}{y}=-\frac{1}{z}\)
Ta lại có : \(\frac{1}{x^3}+\frac{1}{y^3}+\frac{1}{z^3}=\left(\frac{1}{x}+\frac{1}{y}\right)^3-\left(\frac{3}{x^2y}+\frac{3}{xy^2}\right)+\frac{1}{z^3}\)
\(=\left(-\frac{1}{z}\right)^3-\frac{3}{xy}\left(\frac{1}{x}+\frac{1}{y}\right)+\frac{1}{z^3}\)
\(=-\frac{3}{xy}\cdot-\frac{1}{z}\)\(=\frac{3}{xyz}\)
\(\Rightarrow\frac{1}{x^3}+\frac{1}{y^3}+\frac{1}{z^3}=\frac{3}{xyz}\) ( đpcm )
\(\left(x+y+z\right)^2=x^2+y^2+z^2\)
\(\Leftrightarrow xy+yz+zx=0\)
\(\Leftrightarrow\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=0\)
Ta lại co:
\(\frac{1}{x^3}+\frac{1}{y^3}+\frac{1}{z^3}-\frac{3}{xyz}=\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)\left(\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}-\frac{1}{xy}-\frac{1}{yz}-\frac{1}{zx}\right)=0\)
\(\Rightarrow\frac{1}{x^3}+\frac{1}{y^3}+\frac{1}{z^3}=\frac{3}{xyz}\)
Ta có:
\(\left(x+y+z\right)^2=x^2+y^2+z^2\)
\(\Leftrightarrow xy+yz+zx=0\)
\(\Leftrightarrow\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=0\)
Ta lại có:
\(\frac{1}{x^3}+\frac{1}{y^3}+\frac{1}{z^3}-\frac{3}{xyz}=\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)\left(\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}-\frac{1}{xy}-\frac{1}{yz}-\frac{1}{zx}\right)=0\)
\(\Rightarrow\frac{1}{x^3}+\frac{1}{y^3}+\frac{1}{z^3}=\frac{3}{xyz}\)
