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\(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=0\)
<=> \(\frac{yz}{xyz}+\frac{xz}{xyz}+\frac{xy}{xyz}=0\)
<=> \(\frac{yz+xz+xy}{xyz}=0\)
<=> yz + xz + xy = 0
=> (yz)3 + (xz)3 + (xy)3 = 3 . (yz) . (xz) . (xy) = 3x2y2z2
\(K=\left(\frac{yz}{x^2}+\frac{xz}{y^2}+\frac{xy}{z^2}-2\right)^{2017}=\left(\frac{\left(yz\right)^3}{x^2y^2z^2}+\frac{\left(xz\right)^3}{x^2y^2z^2}+\frac{\left(xy\right)^3}{x^2y^2z^2}-2\right)^{2017}=\left(\frac{3x^2y^2z^2}{x^2y^2z^2}-2\right)^{2017}=\left(3-2\right)^{2017}=1^{2017}=1\)
ĐS: 1
\(x^3+y^3+z^3=3xyz\)
\(x^3+y^3+z^3-3xyz=0\)
\(\left(x+y+z\right)\left(x^2+y^2+z^2-xy-xz-yz\right)=0\)
\(x^2+y^2+z^2-xy-xz-yz=0\left(x+y+z\ne0\right)\)
\(2\times\left(x^2+y^2+z^2-xy-xz-yz\right)=0\times2\)
\(2x^2+2y^2+2z^2-2xy-2xz-2yz=0\)
\(x^2-2xy+y^2+x^2-2xz+z^2+y^2-2yz+z^2=0\)
\(\left(x-y\right)^2+\left(x-z\right)^2+\left(y-z\right)^2=0\)
\(\left[\begin{array}{nghiempt}x-y=0\\x-z=0\\y-z=0\end{array}\right.\)
\(\left[\begin{array}{nghiempt}x=y\\x=z\\y=z\end{array}\right.\)
x = y = z
\(P=\left(1+\frac{x}{y}\right)\left(1+\frac{y}{z}\right)\left(1+\frac{x}{z}\right)\)
\(=\left(1+\frac{x}{x}\right)\left(1+\frac{y}{y}\right)\left(1+\frac{z}{z}\right)\)
\(=\left(1+1\right)\left(1+1\right)\left(1+1\right)\)
\(=2^3\)
\(=8\)
a: x-y-z=0
=>x=y+z; y=x-z; z=x-y
\(K=\dfrac{x-z}{x}\cdot\dfrac{y-x}{y}\cdot\dfrac{z+y}{z}=\dfrac{y\cdot\left(-z\right)\cdot x}{xyz}=-1\)
b: Tham khảo:
