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=\(\left(\frac{1}{x\left(x-y\right)}-\frac{3y^2}{x\left(x-y\right)\left(x^2+xy+y^2\right)}-\frac{y}{x\left(x^2+xy+y^2\right)}\right)\)\(\left(\frac{y\left(x+y\right)+x^2}{x+y}\right)\)
=\(\left(\frac{x^2+xy+y^2-3y^2-y\left(x-y\right)}{x\left(x-y\right)\left(x^2+xy+y^2\right)}\right)\) \(\left(\frac{x^2+xy+y^2}{x+y}\right)\)
=\(\left(\frac{x^2+xy-2y^2-xy+y^2}{x\left(x-y\right)}\right)\left(\frac{1}{x+y}\right)\)
=\(\frac{x^2-y^2}{x\left(x-y\right)\left(x+y\right)}\)=\(\frac{\left(x-y\right)\left(x+y\right)}{x\left(x-y\right)\left(x+y\right)}\) =\(\frac{1}{x}\)
\(=\dfrac{a+x+1}{a+x}:\dfrac{a+x-1}{a+x}\cdot\left(\dfrac{2ax-1+a^2+x^2}{2ax}\right)\)
\(=\dfrac{a+x+1}{a+x-1}\cdot\dfrac{\left(a+x\right)^2-1}{2ax}\)
\(=\dfrac{a+x+1}{a+x-1}\cdot\dfrac{\left(a+x+1\right)\left(a+x-1\right)}{2ax}\)
\(=\dfrac{\left(a+x+1\right)^2}{2ax}\)
\(=\dfrac{x^3-1}{x}\cdot\dfrac{x^2-1+x^2+x+1}{\left(x-1\right)\left(x^2+x+1\right)}=\dfrac{2x^2+x}{x}=2x+1\)
\(\frac{2}{xy}:\left(\frac{1}{x}-\frac{1}{y}\right)^2-\frac{x^2+y^2}{\left(x-y\right)^2}\left(ĐK:x\ne0;y\ne0\right)\)
\(=\frac{2}{xy}:\left(\frac{y-x}{xy}\right)^2-\frac{x^2+y^2}{\left(x-y\right)^2}\)
\(=\frac{2}{xy}\cdot\frac{x^2y^2}{\left(y-x\right)^2}-\frac{x^2+y^2}{\left(x-y\right)^2}\)
\(=\frac{-2xy}{\left(x-y\right)^2}+\frac{x^2+y^2}{\left(x-y\right)^2}\)
\(=\frac{-2xy+x^2+y^2}{\left(x-y\right)^2}\)
\(=\frac{\left(x-y\right)^2}{\left(x-y\right)^2}=1\)
\(\frac{2}{xy}:\left(\frac{1}{x}-\frac{1}{y}\right)^2-\frac{x^2+y^2}{\left(x-y\right)^2}\left(dk:x\ne y\ne0\right)\)
miik ko nghĩ nó là toán lớp 7 đâu bn
\(\left[\frac{x}{\left(x+4\right)\left(x-4\right)}-\frac{x-4}{x\left(x+4\right)}\right]:\frac{2\left(x-2\right)}{x\left(x+4\right)}\)\(=\left[\frac{x^2-\left(x-4\right)^2}{x\left(x+4\right)\left(x-4\right)}\right].\left[\frac{x\left(x+4\right)}{2\left(x-2\right)}\right]\)\(=\left(\frac{x^2-x^2+8x-16}{x\left(x+4\right)\left(X-4\right)}\right).\frac{x\left(x+4\right)}{2\left(x-2\right)}=\frac{8\left(x-2\right).x\left(x+4\right)}{x\left(x+4\right)\left(x-4\right).2\left(x-2\right)}=\frac{4}{x-4}\)
\(A=\frac{x^2}{x^2-1}-\frac{x^2}{x^2+1}\left(\frac{x}{x+1}+\frac{1}{x^2+x}\right)\)
=>\(A=\frac{x^2}{\left(x-1\right)\left(x+1\right)}-\frac{x^2}{x^2+1}\left[\frac{x}{x+1}+\frac{1}{x\left(x+1\right)}\right]\)
=>\(A=\frac{x^2}{\left(x-1\right)\left(x+1\right)}-\frac{x^2}{x^2+1}\left[\frac{x^2}{x\left(x+1\right)}+\frac{1}{x\left(x+1\right)}\right]\)
=>\(A=\frac{x^2}{\left(x-1\right)\left(x+1\right)}-\frac{x^2}{x^2+1}.\frac{x^2+1}{x\left(x+1\right)}\)
=>\(A=\frac{x^2}{\left(x-1\right)\left(x+1\right)}-\frac{x}{x+1}\)
=>\(A=\frac{x^2}{\left(x-1\right)\left(x+1\right)}-\frac{x\left(x-1\right)}{\left(x+1\right)\left(x-1\right)}\)
=>\(A=\frac{x^2-x\left(x-1\right)}{\left(x-1\right)\left(x+1\right)}\)
=>\(A=\frac{x^2-x^2+x}{\left(x-1\right)\left(x+1\right)}\)
=>\(A=\frac{x}{x^2-1}\)
- Ta chứng minh bất đẳng thức phụ dưới đây: \(\frac{1}{\sqrt{x}\left(x+1\right)}=\frac{\sqrt{x}}{x\left(x+1\right)}=\sqrt{x}\left(\frac{1}{x}-\frac{1}{x+1}\right)=\sqrt{x}\left(\frac{1}{\sqrt{x}}-\frac{1}{\sqrt{x+1}}\right)\left(\frac{1}{\sqrt{x}}+\frac{1}{\sqrt{x+1}}\right)\)\(=\left(1+\frac{\sqrt{x}}{\sqrt{x+1}}\right)\left(\frac{1}{\sqrt{x}}-\frac{1}{\sqrt{x+1}}\right)< 2\left(\frac{1}{\sqrt{x}}-\frac{1}{\sqrt{x+1}}\right)\)
Áp dụng : \(\frac{1}{\sqrt{1}.2}< 2.\left(1-\frac{1}{\sqrt{2}}\right)\)
\(\frac{1}{\sqrt{2}.3}< 2.\left(\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{3}}\right)\)
...................................
\(\frac{1}{\sqrt{2015}.2016}< 2.\left(\frac{1}{\sqrt{2015}}-\frac{1}{\sqrt{2016}}\right)\)
Cộng các BĐT trên với nhau được : \(\frac{1}{2}+\frac{1}{3\sqrt{2}}+\frac{1}{4\sqrt{3}}+...+\frac{1}{2016\sqrt{2015}}< 2\left(1-\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{3}}+...+\frac{1}{\sqrt{2015}}-\frac{1}{\sqrt{2016}}\right)=2\left(1-\frac{1}{\sqrt{2016}}\right)< 2\left(1-\frac{1}{\sqrt{2025}}\right)=\frac{88}{45}\)
Từ đó suy ra đpcm
Cái ............... là gì vậy bn
\(=\left(\frac{x^3+8}{4x}\right):\left(\frac{x^2-2x+4}{4x}\right)=\frac{\left(x+2\right)\left(x^2-2x+4\right)}{4x}.\frac{4x}{\left(x^2-2x+\right)}=x+2\)