Question

cho \(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}=0\)

chứng minh: \(xyz\left(\dfrac{1}{x^3}+\dfrac{1}{y^3}+\dfrac{1}{z^3}\right)=3\)

Answer 1

Đầu tiên ta cm:\(a+b+c=0\Rightarrow a^3+b^3+c^3=3abc\)

\(\Leftrightarrow a^3+b^3+\left(-a-b\right)^3=3abc\)

\(\Leftrightarrow a^3+b^3-a^3-3a^2b-3ab^2-b^3=3abc\)

\(\Leftrightarrow-3a^2b-3ab^2=3abc\)

\(\Leftrightarrow-3ab\left(a+b\right)=3abc\)

\(\Leftrightarrow-3ab\cdot\left(-c\right)=3abc\)(đúng)

Áp dụng:\(\Rightarrow xyz\cdot\left(\dfrac{1}{x^3}+\dfrac{1}{y^3}+\dfrac{1}{z^3}\right)=xyz\cdot\dfrac{3}{xyz}=3\left(đpcm\right)\)