Rút gọn \(\left[\dfrac{x^2-y^2}{xy}-\dfrac{1}{x+y}\left(\dfrac{x^2}{y}-\dfrac{y^2}{x}\right)\right]:\dfrac{x-y}{x}\)
Những câu hỏi liên quan
Rút gọn biểu thức:
\(A=\left[\dfrac{1}{x^2+2xy+y^2}-\dfrac{1}{x^2-y^2}\right]:\dfrac{4xy}{y^2-x^2}\)
A= \(\left[\dfrac{1}{x^2+2xy+y^2}-\dfrac{1}{x^2-y^2}\right]:\dfrac{4xy}{y^2-x^2}\)
\(=\left[\dfrac{1}{\left(x+y\right)^2}+\dfrac{1}{y^2-x^2}\right]:\dfrac{4xy}{y^2-x^2}\)
=\(\left[\dfrac{1}{\left(x+y\right)^2}+\dfrac{1}{\left(y-x\right)\left(y+x\right)}\right]:\dfrac{4xy}{y^2-x^2}\)
=\(\left[\dfrac{y-x}{\left(x+y\right)^2.\left(y-x\right)}+\dfrac{y+x}{\left(x+y\right)^2\left(y-x\right)}\right]:\dfrac{4xy}{y^2-x^2}\)
=\(\left[\dfrac{y-x+y+x}{\left(x+y\right)^2\left(y-x\right)}\right]:\dfrac{4xy}{y^2-x^2}\)
\(=\dfrac{2y}{\left(x+y\right)^2\left(y-x\right)}:\dfrac{4xy}{y^2-x^2}\)
=\(\dfrac{2y.\left(y-x\right)\left(y+x\right)}{\left(x+y\right)^2\left(y-x\right)4xy}\)
=\(\dfrac{1}{\left(x+y\right)2x}\)
=\(\dfrac{1}{2x^2+2xy}\)
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Bài 2. Cho A=\(\dfrac{\sqrt{x}-\sqrt{y}}{xy\sqrt{xy}}\) :\([\left(\dfrac{1}{x}+\dfrac{1}{y}\right)\dfrac{1}{xy+2\sqrt{xy}}+\dfrac{2}{\left(\sqrt{x}+\sqrt{y}\right)^3}\left(\dfrac{1}{\sqrt{x}}+\dfrac{1}{\sqrt{y}}\right)]\)
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Thực hiện phép tính:
a) dfrac{1}{x-y}+dfrac{3xy}{y^3-x^3}+dfrac{x-y}{x^2+xy+y^2}
b) dfrac{2x+y}{2x^2-xy}+dfrac{16x}{y^2-4x^2}+dfrac{2x-y}{2x^2+xy}
c) dfrac{xy}{ab}+dfrac{left(x-aright)left(y-aright)}{aleft(a-bright)}-dfrac{left(x-bright)left(y-bright)}{bleft(a-bright)}
d) dfrac{x^3}{x-1}-dfrac{x^2}{x+1}-dfrac{1}{x-1}+dfrac{1}{x+1}
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Thực hiện phép tính:
a) \(\dfrac{1}{x-y}+\dfrac{3xy}{y^3-x^3}+\dfrac{x-y}{x^2+xy+y^2}\)
b) \(\dfrac{2x+y}{2x^2-xy}+\dfrac{16x}{y^2-4x^2}+\dfrac{2x-y}{2x^2+xy}\)
c) \(\dfrac{xy}{ab}+\dfrac{\left(x-a\right)\left(y-a\right)}{a\left(a-b\right)}-\dfrac{\left(x-b\right)\left(y-b\right)}{b\left(a-b\right)}\)
d) \(\dfrac{x^3}{x-1}-\dfrac{x^2}{x+1}-\dfrac{1}{x-1}+\dfrac{1}{x+1}\)
a: \(=\dfrac{1}{x-y}-\dfrac{3xy}{\left(x-y\right)\left(x^2+xy+y^2\right)}+\dfrac{x-y}{x^2+xy+y^2}\)
\(=\dfrac{x^2+xy+y^2-3xy+x^2-2xy+y^2}{\left(x-y\right)\left(x^2+xy+y^2\right)}\)
\(=\dfrac{2x^2-4xy+2y^2}{\left(x-y\right)\left(x^2+xy+y^2\right)}=\dfrac{2\left(x-y\right)}{x^2+xy+y^2}\)
d: \(=\dfrac{x^3-1}{x-1}-\dfrac{x^2-1}{x+1}\)
\(=x^2+x+1-x+1=x^2+2\)
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F = \(\dfrac{\sqrt{x}-\sqrt{y}}{xy\sqrt{xy}}:\left[\left(\dfrac{1}{x}+\dfrac{1}{y}\right).\dfrac{1}{x+y+2\sqrt{xy}}+\dfrac{2}{\left(\sqrt{x}+\sqrt{y}\right)^3}.\left(\dfrac{1}{\sqrt{x}}+\dfrac{1}{\sqrt{y}}\right)\right]\)
\(F=\dfrac{\sqrt{x}-\sqrt{y}}{xy\sqrt{xy}}:\left[\dfrac{x+y}{xy}\cdot\dfrac{1}{\left(\sqrt{x}+\sqrt{y}\right)^2}+\dfrac{2}{\sqrt{xy}\left(\sqrt{x}+\sqrt{y}\right)^2}\right]\)
\(=\dfrac{\sqrt{x}-\sqrt{y}}{xy\sqrt{xy}}:\left[\dfrac{x+y+2\sqrt{xy}}{xy\left(\sqrt{x}+\sqrt{y}\right)^2}\right]\)
\(=\dfrac{\sqrt{x}-\sqrt{y}}{xy\sqrt{xy}}\cdot xy=\dfrac{\sqrt{x}-\sqrt{y}}{\sqrt{xy}}\)
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\(C=\dfrac{\sqrt{x}-\sqrt{y}}{xy\sqrt{xy}}:\left(\dfrac{1}{x}+\dfrac{1}{y}\right).\dfrac{1}{x+y+2\sqrt{xy}}+\dfrac{2}{\left(\sqrt{x}+\sqrt{y}\right)^3}.\left(\dfrac{1}{\sqrt{x}}+\dfrac{1}{\sqrt{y}}\right)\)
a) Rút gọn
b) Tính C với x=2-\(\sqrt{3}\); y=2+\(\sqrt{3}\)
Cho x,y,z>0. CM: \(\dfrac{xy}{z^2\left(x+y\right)}+\dfrac{yz}{x^2\left(y+z\right)}+\dfrac{zx}{y^2\left(z+x\right)}\ge\dfrac{1}{2}\left(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}\right)\)
\(VT=\dfrac{\left(\dfrac{1}{z}\right)^2}{\dfrac{1}{x}+\dfrac{1}{y}}+\dfrac{\left(\dfrac{1}{x}\right)^2}{\dfrac{1}{y}+\dfrac{1}{z}}+\dfrac{\left(\dfrac{1}{y}\right)^2}{\dfrac{1}{x}+\dfrac{1}{z}}\ge\dfrac{\left(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}\right)^2}{2\left(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}\right)}=\dfrac{1}{2}\left(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}\right)\)
Dâu "=" xảy ra khi \(x=y=z\)
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Tính:
a) dfrac{x^2}{left(x-yright)left(x-zright)}+dfrac{y^2}{left(y-zright)left(y-xright)}+dfrac{z^2}{left(z-xright)left(z-yright)}
b) dfrac{x^2-yz}{left(x+yright)left(x+zright)}+dfrac{y^2-zx}{left(y+zright)left(y+xright)}+dfrac{z^2-xy}{left(z+xright)left(z+yright)}
c) dfrac{1}{xleft(x-yright)left(x-zright)}+dfrac{1}{yleft(y-xright)left(y-zright)}+dfrac{1}{zleft(z-xright)left(z-yright)}
d) dfrac{1}{xleft(x+1right)}+dfrac{1}{left(x+1right)left(x+2right)}+dfrac{1}{left(x+2right)left(x+3right)}...
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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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d)
\(\dfrac{1}{x\left(x+1\right)}+\dfrac{1}{\left(x+1\right)\left(x+2\right)}+\dfrac{1}{\left(x+2\right)\left(x+3\right)}+.....+\dfrac{1}{\left(x+99\right)\left(x+100\right)}\)=\(\dfrac{1}{x}-\dfrac{1}{x+1}+\dfrac{1}{x+1}-\dfrac{1}{x+2}+\dfrac{1}{x+2}-\dfrac{1}{x+3}+.....-\dfrac{1}{x+99}+\dfrac{1}{x+100}\)=\(\dfrac{1}{x}-\dfrac{1}{x+100}\)
=\(\dfrac{x+100}{x\left(x+100\right)}-\dfrac{x}{x\left(x+100\right)}\)
=\(\dfrac{x+100-x}{x\left(x+100\right)}=\dfrac{100}{x\left(x+100\right)}\)
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thuc hien phep tinh
a.left(dfrac{2x+1}{2x-1}-dfrac{2x-1}{2x+1}right):dfrac{4x}{10x-5}
b.left(dfrac{1}{x^2+1}-dfrac{2-x}{x+1}right):left(dfrac{1}{x}+1-2right)
c.dfrac{1}{x-1}-dfrac{x^3-x}{x^2+1}.left(dfrac{1}{x^2-2x+1}+dfrac{1}{1-x^2}right)
d.left(dfrac{x^2+xy}{x^3+x^2y+xy^2+y^3}+dfrac{y}{x^2+y^2}right):left(dfrac{1}{x-y}-dfrac{2xy}{x^3-x^2y+xy^2-y^3}right)
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thuc hien phep tinh
a.\(\left(\dfrac{2x+1}{2x-1}-\dfrac{2x-1}{2x+1}\right):\dfrac{4x}{10x-5}\)
b.\(\left(\dfrac{1}{x^2+1}-\dfrac{2-x}{x+1}\right):\left(\dfrac{1}{x}+1-2\right)\)
c.\(\dfrac{1}{x-1}-\dfrac{x^3-x}{x^2+1}.\left(\dfrac{1}{x^2-2x+1}+\dfrac{1}{1-x^2}\right)\)
d.\(\left(\dfrac{x^2+xy}{x^3+x^2y+xy^2+y^3}+\dfrac{y}{x^2+y^2}\right):\left(\dfrac{1}{x-y}-\dfrac{2xy}{x^3-x^2y+xy^2-y^3}\right)\)
a: \(=\dfrac{4x^2+4x+1-\left(4x^2-4x+1\right)}{\left(2x-1\right)\left(2x+1\right)}\cdot\dfrac{5\left(2x-1\right)}{4x}\)
\(=\dfrac{8x}{2x+1}\cdot\dfrac{5}{4x}=\dfrac{10}{2x+1}\)
c: \(=\dfrac{1}{x-1}-\dfrac{x\left(x-1\right)\left(x+1\right)}{x^2+1}\cdot\left(\dfrac{x+1-x+1}{\left(x-1\right)^2\cdot\left(x+1\right)}\right)\)
\(=\dfrac{1}{x-1}-\dfrac{x}{x^2+1}\cdot\dfrac{2}{\left(x-1\right)}=\dfrac{x^2+1-2x}{\left(x-1\right)\left(x^2+1\right)}=\dfrac{x-1}{x^2+1}\)
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\(\dfrac{y}{2x^2-xy}+\dfrac{4x}{y^2-2xy}\)
\(\dfrac{1}{x+2}+\dfrac{3}{x^2-4}+\dfrac{x-14}{\left(x^2+4x+4\right).\left(x-2\right)}\)
\(\dfrac{1}{x+2}+\dfrac{1}{\left(x+2\right).\left(4x+7\right)}\)
\(\dfrac{1}{x+3}+\dfrac{1}{\left(x+3\right).\left(x+2\right)}+\dfrac{1}{\left(x+2\right).\left(4x+7\right)}\)
\(\left(1\right)=\dfrac{y}{x\left(2x-y\right)}-\dfrac{4x}{y\left(2x-y\right)}=\dfrac{y^2-4x^2}{xy\left(2x-y\right)}=\dfrac{-\left(y-2x\right)\left(y+2x\right)}{xy\left(y-2x\right)}=\dfrac{-y-2x}{xy}\\ \left(2\right)=\dfrac{x^2-4+3x+6+x-14}{\left(x+2\right)^2\left(x-2\right)}=\dfrac{x^2+4x-12}{\left(x+2\right)^2\left(x-2\right)}=\dfrac{\left(x-2\right)\left(x+6\right)}{\left(x+2\right)^2\left(x-2\right)}=\dfrac{x+6}{\left(x+2\right)^2}\\ \left(3\right)=\dfrac{4\left(x+2\right)}{\left(x+2\right)\left(4x+7\right)}=\dfrac{4}{4x+7}\\ \left(4\right)=\dfrac{4x^2+15x+4+4x+7+1}{\left(x+2\right)\left(x+3\right)\left(4x+7\right)}=\dfrac{4x^2+19x+12}{\left(x+2\right)\left(x+3\right)\left(4x+7\right)}\)
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