Ta có :
\(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=0\)
\(\Leftrightarrow\)\(\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)^3=0^3\)
\(\Leftrightarrow\)\(\left(\frac{1}{x}\right)^3+\left(\frac{1}{y}\right)^3+\left(\frac{1}{z}\right)^3+3\left(\frac{1}{x}+\frac{1}{y}\right)\left(\frac{1}{y}+\frac{1}{z}\right)\left(\frac{1}{z}+\frac{1}{x}\right)=0\)
\(\Leftrightarrow\)\(\frac{1^3}{x^3}+\frac{1^3}{y^3}+\frac{1^3}{z^3}=-3\left(\frac{1}{x}+\frac{1}{y}\right)\left(\frac{1}{y}+\frac{1}{z}\right)\left(\frac{1}{z}+\frac{1}{x}\right)\)
Lại có :
\(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=0\)
\(\Rightarrow\)\(\hept{\begin{cases}\frac{1}{x}+\frac{1}{y}=\frac{-1}{z}\\\frac{1}{y}+\frac{1}{z}=\frac{-1}{x}\\\frac{1}{z}+\frac{1}{x}=\frac{-1}{y}\end{cases}}\)
\(\Leftrightarrow\)\(\frac{1}{x^3}+\frac{1}{y^3}+\frac{1}{z^3}=\left(-3\right).\frac{-1}{z}.\frac{-1}{x}.\frac{-1}{y}\)
\(\Leftrightarrow\)\(\frac{1}{x^3}+\frac{1}{y^3}+\frac{1}{z^3}=\frac{3}{xyz}\) ( đpcm )
Vậy nếu \(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=0\) thì \(\frac{1}{x^3}+\frac{1}{y^3}+\frac{1}{z^3}=\frac{3}{xyz}\)
Chúc bạn học tốt ~
\(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=0\Leftrightarrow\frac{1}{x}+\frac{1}{y}=\frac{-1}{z}\)
\(\Rightarrow\left(\frac{1}{x}+\frac{1}{y}\right)^3=\left(-\frac{1}{z}\right)^3\Leftrightarrow\frac{1}{x^3}+\frac{1}{y^3}+\frac{3}{x^2y}+\frac{3}{xy^2}=-\frac{1}{z^3}\)
\(\Leftrightarrow\frac{1}{x^3}+\frac{1}{y^3}+\frac{1}{z^3}=\frac{-3}{x^2y}-\frac{3}{xy^2}=\frac{-3}{xy}.\left(\frac{1}{x}+\frac{1}{y}\right)=\frac{-3}{xy}.-\frac{1}{z}=\frac{3}{xyz}\)
\(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=0\Rightarrow\frac{1}{x}+\frac{1}{y}=-\frac{1}{z};\frac{1}{x}+\frac{1}{z}=-\frac{1}{y};\frac{1}{y}+\frac{1}{z}=-\frac{1}{x}\)
\(2\left(\frac{1}{x^3}+\frac{1}{y^3}+\frac{1}{z^3}\right)=\left(\frac{1}{x^3}+\frac{1}{y^3}\right)+\left(\frac{1}{x^3}+\frac{1}{z^3}\right)+\left(\frac{1}{y^3}+\frac{1}{z^3}\right)\)
\(=\left(\frac{1}{x}+\frac{1}{y}\right)\left(\frac{1}{x^2}-\frac{1}{xy}+\frac{1}{y^2}\right)+\left(\frac{1}{x}+\frac{1}{z}\right)\left(\frac{1}{x^2}-\frac{1}{xz}+\frac{1}{z^2}\right)+\left(\frac{1}{y}+\frac{1}{z}\right)\left(\frac{1}{y^2}-\frac{1}{yz}+\frac{1}{z^2}\right)\)
\(=-\frac{1}{z}\left(\frac{1}{x^2}-\frac{1}{xy}+\frac{1}{y^2}\right)-\frac{1}{y}\left(\frac{1}{x^2}-\frac{1}{xz}+\frac{1}{z^2}\right)-\frac{1}{x}\left(\frac{1}{y^2}-\frac{1}{yz}+\frac{1}{z^2}\right)\)
\(=-\frac{1}{x^2z}+\frac{1}{xyz}-\frac{1}{y^2z}-\frac{1}{x^2y}+\frac{1}{xyz}-\frac{1}{yz^2}-\frac{1}{xy^2}+\frac{1}{xyz}-\frac{1}{xz^2}\)
\(=\left(-\frac{1}{x^2z}-\frac{1}{x^2y}\right)+\left(-\frac{1}{xy^2}-\frac{1}{y^2z}\right)+\left(-\frac{1}{xz^2}-\frac{1}{yz^2}\right)+\frac{3}{xyz}\)
\(=-\frac{1}{x^2}\left(\frac{1}{z}+\frac{1}{y}\right)-\frac{1}{y^2}\left(\frac{1}{x}+\frac{1}{z}\right)-\frac{1}{z^2}\left(\frac{1}{x}+\frac{1}{y}\right)+\frac{3}{xyz}\)
\(=-\frac{1}{x^2}\cdot-\frac{1}{x}+-\frac{1}{y^2}\cdot-\frac{1}{y}+-\frac{1}{z^2}\cdot-\frac{1}{z}+\frac{3}{xyz}=\frac{1}{x^3}+\frac{1}{y^3}+\frac{1}{z^3}+\frac{3}{xyz}\)
\(\Rightarrow2\left(\frac{1}{x^3}+\frac{1}{y^3}+\frac{1}{z^3}\right)=\frac{1}{x^3}+\frac{1}{y^3}+\frac{1}{z^3}+\frac{3}{xyz}\Rightarrow\frac{1}{x^3}+\frac{1}{y^3}+\frac{1}{z^3}=\frac{3}{xyz}\)(đpcm)