1/ CMR : \(\frac{2011^3+11^3}{2011^3+2000^3}=\frac{2011+11}{2011+2000}\)
2/ Xét \(A=\left(\frac{a+1}{ab+1}+\frac{ab+a}{ab-1}-1\right):\left(\frac{a+1}{ab+1}-\frac{ab+a}{ab-1}+1\right)\)
a/ rút gọn
b/ tìm GTNN mà A đạt được biết a + b = 4
3/ CMR giá trị biểu thức biểnsau ko phụ thuộc vào giá trị của biến
\(\frac{2}{xy}:\left(\frac{1}{x}-\frac{1}{y}\right)^2-\frac{x^2+y^2}{\left(x-y\right)^2}\) khi \(x\ne0;y\ne0;x\ne y\)
\(3,\frac{2}{xy}:\left(\frac{1}{x}-\frac{1}{y}\right)^2-\frac{x^2+y^2}{\left(x-y\right)^2}\)
\(=\frac{2}{xy}:\left[\left(\frac{1}{x}\right)^2-2.\frac{1}{x}.\frac{1}{y}+\left(\frac{1}{y}\right)^2\right]-\frac{x^2+y^2}{\left(x-y\right)^2}\)
\(=\frac{2}{xy}:\left[\frac{1}{x^2}-\frac{2}{xy}+\frac{1}{y^2}\right]-\frac{x^2+y^2}{x^2-2xy+y^2}\)
\(=\frac{2}{xy}:\left[\frac{y^2-2.xy+x^2}{x^2y^2}\right]-\frac{x^2+y^2}{\left(x-y\right)^2}\)
\(=\frac{2}{xy}.\frac{x^2y^2}{x^2-2xy+y^2}-\frac{x^2+y^2}{x^2-2xy+y^2}\)
\(=\frac{2xy}{x^2-2xy+y^2}+\frac{-x^2-y^2}{x^2-2xy-y^2}\)
\(=\frac{2xy-x^2-y^2}{x^2-2xy+y^2}=\frac{-\left(x^2-2xy+y^2\right)}{x^2-2xy+y^2}=-1\)
\(\frac{2011^3+11^3}{2011^3+2000^3}\)
\(=\frac{\left(2011+11\right)\left(2011^2-2011.11+11^2\right)}{\left(2011+2000\right)\left(2011^2-2011.2000+2000^2\right)}\)
\(=\frac{\left(2011+11\right)\left[2011^2-11\left(2011-11\right)\right]}{\left(2011+2000\right)\left[2011^2-2000\left(2011-2000\right)\right]}\)
\(=\frac{\left(2011+11\right)\left(2011^2-11.2000\right)}{\left(2011+2000\right)\left(2011^2-2000.11\right)}\)
\(=\frac{2011+11}{2011+2000}\left(2011^2-11.2000\ne0\right)\)
đpcm
\(A=\left(\frac{a+1}{ab+1}+\frac{ab+a}{ab-1}-1\right):\left(\frac{a+1}{ab+1}-\frac{ab+a}{ab-1}+1\right)\)
\(A=\left[\frac{\left(a+1\right)\left(ab-1\right)+\left(ab+a\right)\left(ab+1\right)-\left(ab+1\right)\left(ab-1\right)}{\left(ab+1\right)\left(ab-1\right)}\right]:\left[\frac{\left(a+1\right)\left(ab-1\right)-\left(ab+a\right)\left(ab+1\right)+\left(ab+1\right)\left(ab-1\right)}{\left(ab+1\right)\left(ab-1\right)}\right]\)\(A=\left[\frac{a^2b-a+ab-1+a^2b^2+ab+a^2b+a-a^2b^2+1}{\left(ab+1\right)\left(ab-1\right)}\right]:\left[\frac{a^2b-a+ab-1-a^2b^2-ab-a^2b-a+a^2b^2-1}{\left(ab+1\right)\left(ab-1\right)}\right]\)\(A=\left[\frac{2a^2b+2ab}{\left(ab+1\right)\left(ab-1\right)}\right]:\left[\frac{2a^2b-2a}{\left(ab+1\right)\left(ab-1\right)}\right]\)
\(A=\left[\frac{2ab\left(a+1\right)}{\left(ab+1\right)\left(ab-1\right)}\right]:\left[\frac{2a\left(ab-1\right)}{\left(ab+1\right)\left(ab-1\right)}\right]\)
\(A=\left[\frac{2ab\left(a+1\right)}{\left(ab+1\right)\left(ab-1\right)}\right]:\left[\frac{2a}{\left(ab+1\right)}\right]\left(ab-1\ne0\right)\)
\(A=\frac{b\left(a+1\right)}{ab-1}\left(ab+1\ne0;2a\ne0\right)\)
