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\(\frac{x+\frac{1}{y}}{y+\frac{1}{x}}=\frac{\frac{xy}{y}}{\frac{xy}{x}}=\frac{xy}{y}.\frac{x}{xy}=\frac{x}{y}\)
\(\frac{x+\frac{1}{y}}{y+\frac{1}{x}}=\left(x+\frac{1}{y}\right):\left(y+\frac{1}{x}\right)=\frac{xy+1}{y}:\frac{xy+1}{x}=\frac{\left(xy+1\right)\cdot x}{\left(xy+1\right)\cdot y}=\frac{x}{y}\).
a/ \(\frac{7x-14y}{x^2-4y^2}=\frac{7\left(x-2y\right)}{x^2-\left(2y\right)^2}=\frac{7\left(x-2y\right)}{\left(x-2y\right)\left(x+2y\right)}=\frac{7}{x+2y}.\)
b/ \(\frac{1-\frac{2y}{x}+\frac{y^2}{x^2}}{\frac{1}{x}-\frac{1}{y}}=\frac{\frac{x^2-2xy+y^2}{x^2}}{\frac{y-x}{xy}}=\frac{\left(x-y\right)^2}{x^2}.\frac{xy}{-\left(x-y\right)}=-\frac{y\left(x-y\right)}{x}=\frac{y\left(y-x\right)}{x}\)
\(\left(\frac{x}{x-1}-\frac{x+1}{x}\right):\left(\frac{x}{x+1}-\frac{x-1}{x}\right)\)
\(=\left(\frac{x^2-\left(x-1\right)\left(x+1\right)}{\left(x-1\right).x}\right):\left(\frac{x^2-\left(x-1\right)\left(x+1\right)}{x\left(x+1\right)}\right)\)
\(=\frac{x^2-\left(x-1\right)\left(x+1\right)}{x\left(x-1\right)}.\frac{x\left(x+1\right)}{x^2-\left(x-1\right)\left(x+1\right)}\)
\(=\frac{x+1}{x-1}\)
Ta có : \(\frac{\frac{1}{1-x}+\frac{1}{1+x}}{\frac{1}{1-x}-\frac{1}{1+x}}=\frac{\frac{1+x}{1-x^2}+\frac{1-x}{1-x^2}}{\frac{1+x}{1-x^2}-\frac{1-x}{1-x^2}}=\frac{\frac{1+x+1-x}{1-x^2}}{\frac{1+x-1+x}{1-x^2}}\)
\(=\frac{\frac{2}{1-x^2}}{\frac{2x}{1-x^2}}=\left(\frac{2}{1-x^2}\right)\left(\frac{1-x^2}{2x}\right)=\frac{2}{2x}=\frac{1}{x}\)
\(\frac{\frac{1}{1-x}+\frac{1}{1+x}}{\frac{1}{1-x}-\frac{1}{1+x}}< =>\left(\frac{1}{1-x}+\frac{1}{1+x}\right):\left(\frac{1}{1-x}-\frac{1}{1+x}\right)\)
<=> tự làm nha