Ta có :
\(\left(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}\right)^2=\dfrac{1}{x^2}+\dfrac{1}{y^2}+\dfrac{1}{z^2}+2\left(\dfrac{1}{xy}+\dfrac{1}{yz}+\dfrac{1}{xz}\right)\)\(=\dfrac{1}{x^2}+\dfrac{1}{y^2}+\dfrac{1}{z^2}+2\left(\dfrac{z+x+y}{xyz}\right)\)
\(=\dfrac{1}{x^2}+\dfrac{1}{y^2}+\dfrac{1}{z^2}+2\left(\dfrac{0}{xyz}\right)\)
\(=\dfrac{1}{x^2}+\dfrac{1}{y^2}+\dfrac{1}{z^2}\)
Chứng minh đẳng thức:
a) \(\dfrac{y}{\left(x-y\right)\left(y-z\right)}+\dfrac{z}{\left(y-z\right)\left(z-x\right)}+\dfrac{x}{\left(z-x\right)\left(x-y\right)=0}\)
b) \(\dfrac{x^2}{\left(x-y\right)\left(y-z\right)}+\dfrac{y^2}{\left(y-z\right)\left(y-x\right)}+\dfrac{z^2}{\left(z-x\right)\left(z-y\right)=1}\)
c) \(\dfrac{1}{x\left(x-y\right)\left(x-z\right)}+\dfrac{1}{y\left(y-z\right)\left(y-x\right)}+\dfrac{1}{z\left(z-x\right)\left(z-y\right)}=\dfrac{1}{xyz}\)
Tính:
a) \(\dfrac{x^2}{\left(x-y\right)\left(x-z\right)}+\dfrac{y^2}{\left(y-z\right)\left(y-x\right)}+\dfrac{z^2}{\left(z-x\right)\left(z-y\right)}\)
b) \(\dfrac{x^2-yz}{\left(x+y\right)\left(x+z\right)}+\dfrac{y^2-zx}{\left(y+z\right)\left(y+x\right)}+\dfrac{z^2-xy}{\left(z+x\right)\left(z+y\right)}\)
c) \(\dfrac{1}{x\left(x-y\right)\left(x-z\right)}+\dfrac{1}{y\left(y-x\right)\left(y-z\right)}+\dfrac{1}{z\left(z-x\right)\left(z-y\right)}\)
d) \(\dfrac{1}{x\left(x+1\right)}+\dfrac{1}{\left(x+1\right)\left(x+2\right)}+\dfrac{1}{\left(x+2\right)\left(x+3\right)}+...+\dfrac{1}{\left(x+99\right)\left(x+100\right)}\)
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Tính
1. \(\dfrac{1}{\left(x-y\right)\left(y-z\right)}+\dfrac{1}{\left(y-z\right)\left(z-1\right)}+\dfrac{1}{\left(z-x\right)\left(x-y\right)}\)
2. \(\dfrac{x}{x-2y}+\dfrac{x}{x+2y}+\dfrac{4xy}{4y^2-x}\)
Cộng các phân thức :
a) \(\dfrac{1}{\left(x-y\right)\left(y-z\right)}+\dfrac{1}{\left(y-z\right)\left(z-x\right)}+\dfrac{1}{\left(z-x\right)\left(x-y\right)}\)
b) \(\dfrac{4}{\left(y-x\right)\left(z-x\right)}+\dfrac{3}{\left(y-x\right)\left(y-z\right)}+\dfrac{3}{\left(y-z\right)\left(x-z\right)}\)
c) \(\dfrac{1}{x\left(x-y\right)\left(x-z\right)}+\dfrac{1}{y\left(y-z\right)\left(y-x\right)}+\dfrac{1}{z\left(z-x\right)\left(z-y\right)}\)
Tính:
\(D=\dfrac{4x^2-1}{\left(x-y\right)\cdot\left(x+y\right)}+\dfrac{4y^2-1}{\left(y-z\right)\cdot\left(y-x\right)}+\dfrac{4z^2-1}{\left(z-x\right)\cdot\left(z-y\right)}\)
\(\dfrac{1}{\left(x-y\right)\left(y-z\right)}+\dfrac{1}{\left(y-z\right)\left(z-x\right)}+\dfrac{1}{\left(z-x\right)\left(x-y\right)}\)
Tính:
a) \(\dfrac{x}{\left(x-y\right)\left(x-z\right)}+\dfrac{y}{\left(y-x\right)\left(y-z\right)}+\dfrac{z}{\left(z-x\right)\left(z-y\right)}\)
b) \(\dfrac{x^2}{\left(x-y\right)\left(x-z\right)}+\dfrac{y^2}{\left(y-x\right)\left(y-z\right)}+\dfrac{z^2}{\left(z-x\right)\left(z-y\right)}\)
1)Tìm x,y,z biết : x2+y2+z2+\(\dfrac{1}{x^2}+\dfrac{1}{y^2}\)=4
2)Cho \(\dfrac{x}{a}+\dfrac{y}{b}+\dfrac{z}{c}=0\)và \(\dfrac{a}{x}+\dfrac{b}{y}+\dfrac{c}{z}=2\)
Tính \(\dfrac{a^2}{x^2}+\dfrac{b^2}{y^2}+\dfrac{c^2}{z^2}\)
3)Cho \(x=\dfrac{a}{3a+2}\).Rút gọn biểu thức:
A=\(\dfrac{x+3a}{2-x}+\dfrac{x-3a}{2+x}-\dfrac{2a}{4-x^2}+a\)
Cho x;y;z khác 0 và x+y khác z và y+z khác x thỏa mãn:
\(\dfrac{x^2+y^2-z^2}{2xy}-\dfrac{y^2+z^2-x^2}{2yz}+\dfrac{z^2+x^2-y^2}{2xz}=1\)
Tính P = x + y + z
