rút gọn Q=\(\dfrac{x^2}{xy+y^2}+\dfrac{y^2}{xy-y^2}+\dfrac{x^2+y^2}{xy}\)
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Rút gọn biểu thức:
\(\dfrac{x^2+xy}{x^2+xy+y^2}\) - [\(\dfrac{x\left(2x^2+xy-y^2\right)}{x^3-y^3}\) - 2 + \(\dfrac{y}{y-x}\)] : \(\dfrac{x-y}{x}\) - \(\dfrac{x}{x-y}\)
Ta có: \(\dfrac{x^2+xy}{x^2+xy+y^2}-\left(\dfrac{x\left(2x^2+xy-y^2\right)}{x^3-y^3}-2+\dfrac{y}{y-x}\right):\dfrac{x-y}{x}-\dfrac{x}{x-y}\)
\(=\dfrac{x^2+xy}{x^2+xy+y^2}-\left(\dfrac{x\left(2x^2+xy-y^2\right)}{\left(x-y\right)\left(x^2+xy+y^2\right)}-\dfrac{2\left(x^3-y^3\right)-y\left(x^2+xy+y^2\right)}{\left(x-y\right)\left(x^2+xy+y^2\right)}\right):\dfrac{x-y}{x}-\dfrac{x}{x-y}\)
\(=\dfrac{x^2+xy}{x^2+xy+y^2}-\dfrac{2x^3+x^2y-xy^2-2x^3+2y^3-x^2y-xy^2-y^3}{\left(x-y\right)\left(x^2+xy+y^2\right)}:\dfrac{x-y}{x}-\dfrac{x}{x-y}\)
\(=\dfrac{x\left(x+y\right)}{x^2+xy+y^2}-\dfrac{y^3-2xy^2}{\left(x-y\right)\left(x^2+xy+y^2\right)}:\dfrac{x-y}{x}-\dfrac{x}{x-y}\)
\(=\dfrac{x\left(x+y\right)}{x^2+xy+y^2}+\dfrac{y^2\left(x-y\right)}{\left(x-y\right)\left(x^2+xy+y^2\right)}\cdot\dfrac{x}{x-y}-\dfrac{x}{x-y}\)
\(=\dfrac{x\left(x+y\right)}{x^2+xy+y^2}+\dfrac{xy^2}{\left(x-y\right)\left(x^2+xy+y^2\right)}-\dfrac{x}{x-y}\)
\(=\dfrac{x\left(x^2-y^2\right)}{\left(x-y\right)\left(x^2+xy+y^2\right)}+\dfrac{xy^2}{\left(x-y\right)\left(x^2+xy+y^2\right)}-\dfrac{x\left(x^2+xy+y^2\right)}{\left(x-y\right)\left(x^2+xy+y^2\right)}\)
\(=\dfrac{x^3-xy^2+xy^2-x^3-x^2y-xy^2}{\left(x-y\right)\left(x^2+xy+y^2\right)}\)
\(=\dfrac{-x^2y-xy^2}{\left(x-y\right)\left(x^2+xy+y^2\right)}\)
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Rút gọn:
\(\left(\dfrac{x^2}{x+y}+y\right).\left(\dfrac{1}{x^2-xy}-\dfrac{3y^2}{x^4-xy^3}-\dfrac{y}{x^3+x^2y+xy^2}\right)\)
\(=\frac{x^2+xy+y^2}{x+y}.\left(\frac{1}{\left(x-y\right)x}-\frac{3y^2}{x\left(x^3-y^3\right)}-\frac{y}{x\left(x^2+xy+y^2\right)}\right)\)
\(=\frac{x^2+xy+y^2}{x+y}.\frac{x^2+xy+y^2-3y^2-xy+y^2}{x\left(x-y\right)\left(x^2+xy+y^2\right)}\)
\(=\frac{x^2-y^2}{x\left(x-y\right)\left(x+y\right)}=\frac{\left(x-y\right)\left(x+y\right)}{x\left(x-y\right)\left(x+y\right)}=\frac{1}{x}\)
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Thực hiện phép tính , rút gọn bt
\(\dfrac{2x+y}{2x^2-xy}+\dfrac{16x}{y^2-4x^2}+\dfrac{2x-y}{2x^2+xy}\)
\(\dfrac{x+y}{2\left(x-y\right)}+\dfrac{2}{x^2+3}+\dfrac{1}{x+1}\)
Rút gọn:
\(\dfrac{2}{x}-\left(\dfrac{x^2}{x^2+xy}-\dfrac{x^2-y^2}{xy}-\dfrac{y^2}{xy+y^2}\right):\dfrac{x^3-y^3}{x^2-y^2}\)
\(=\dfrac{2}{x}-\left(\dfrac{x^2}{x\left(x+y\right)}-\dfrac{x^2-y^2}{xy}-\dfrac{y^2}{y\left(x+y\right)}\right):\dfrac{x^3-y^3}{x^2-y^2}\)
\(=\dfrac{2}{x}-\left(\dfrac{x^2y-\left(x^2-y^2\right)\left(x+y\right)-y^2x}{xy\left(x+y\right)}\right)\cdot\dfrac{x+y}{x^2+xy+y^2}\)
\(=\dfrac{2}{x}-\dfrac{x^2y-x^3-x^2y+xy^2+y^3-xy^2}{xy}\cdot\dfrac{1}{x^2+xy+y^2}\)
\(=\dfrac{2}{x}-\dfrac{-\left(x-y\right)\left(x^2+xy+y^2\right)}{xy}\cdot\dfrac{1}{x^2+xy+y^2}\)
\(=\dfrac{2}{x}+\dfrac{x-y}{xy}=\dfrac{y+x-y}{xy}=\dfrac{1}{y}\)
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Rút gọn biểu thức:
\(a,\left(\dfrac{x}{xy-y^2}+\dfrac{2x-y}{xy-x^2}\right):\left(\dfrac{1}{x}+\dfrac{1}{y}\right)\)
\(b,\left(\dfrac{x+y}{2x-2y}-\dfrac{x-y}{2x+2y}-\dfrac{2y^2}{y-x}\right):\dfrac{2y}{x-y}\)
Rút gọn biểu thức:
\(a,\left(\dfrac{x}{xy-y^2}+\dfrac{2x-y}{xy-x^2}\right):\left(\dfrac{1}{x}+\dfrac{1}{y}\right)\)
\(b,\left(\dfrac{x+y}{2x-2y}-\dfrac{x-y}{2x+2y}-\dfrac{2y^2}{y-x}\right):\dfrac{2y}{x-y}\)
\(a,\frac{x}{xy-y^2}+\frac{2x-y}{xy-x^2}:\left(\frac{1}{x}+\frac{1}{y}\right)\)
\(=\left(\frac{x}{y\left(x-y\right)}+\frac{y-2x}{x\left(x-y\right)}\right):\left(\frac{y}{xy}+\frac{x}{xy}\right)\)
\(=\left(\frac{x-y}{x\left(x-y\right)}\right):\left(\frac{x+y}{xy}\right)\)
\(=\frac{1}{x}.\frac{xy}{x+y}=\frac{y}{x+y}\)
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9 tháng 7 2021 lúc 16:25
\(P=\left(\sqrt{x}+\dfrac{y-\sqrt{xy}}{\sqrt{x}+\sqrt{y}}\right)\):\(\left(\dfrac{x}{\sqrt{xy}+y}+\dfrac{y}{\sqrt{xy}-x}-\dfrac{x+y}{\sqrt{xy}}\right)\)
a) Với giá trị nào cùa x thì biểu thức có nghĩa
b) Rút gọn P
c) Tím P với x=3 và y=\(\dfrac{2}{2-\sqrt{3}}\)
Giúp với ạ
Cho biểu thức:
A = (\(\sqrt{x}\) + \(\dfrac{y-\sqrt{xy}}{\sqrt{x}+\sqrt{y}}\)) : (\(\dfrac{x}{\sqrt{xy}+y}\) + \(\dfrac{y}{\sqrt{xy}-x}\) - \(\dfrac{x+y}{\sqrt{xy}}\))
a) Rút gọn A
b) Tính giá trị của biểu thức A biết x = 3; y = 4 + 2\(\sqrt{3}\)