Question

cho x,y,z\(\in\)R thỏa mãn \(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{1}{x+y+z}\)

Hãy tính giá trị của biểu thức : M=  \(\frac{3}{4}+\left(x^8-y^8\right)\left(y^9+z^9\right)\left(z^{10}-x^{10}\right)\)

Answer 1

\(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{1}{x+y+z}\Leftrightarrow\frac{xy+yz+zx}{xyz}=\frac{1}{x+y+z}\)

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

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

\(\Leftrightarrow x=-y\text{ hoặc }y=-z\text{ hoặc }z=-x\)

\(+\text{Nếu }x=-y\text{ thì }x^8=\left(-y\right)^8=y^8\Rightarrow x^8-y^8=0\Rightarrow M=\frac{3}{4}\)

\(+\text{Nếu }y=-z\text{ thì }y^9=\left(-z\right)^9=-z^9\Rightarrow y^9+z^9=0\Rightarrow M=\frac{3}{4}\)

\(+\text{Nếu }z=-x\text{ thì }z^{10}=\left(-x\right)^{10}=x^{10}\Rightarrow z^{10}-x^{10}=0\Rightarrow M=\frac{3}{4}\)

\(\text{Vậy M}=\frac{3}{4}.\)

 

Answer 2

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