`1)A=x/(x-4)+1/(\sqrt{x}-2)+1/(\sqrt{x}+2)(x>=0,x\ne4)`
`=x/((\sqrt{x}+2)(\sqrt{x}-2))+1/(\sqrt{x}-2)+1/(\sqrt{x}+2)`
`=x/((\sqrt{x}+2)(\sqrt{x}-2))+(\sqrt{x}+2)/((\sqrt{x}+2)(\sqrt{x}-2))+(\sqrt{x}-2)/((\sqrt{x}+2)(\sqrt{x}-2))`
`=(x+\sqrt{x}+2+\sqrt{x}-2)/((\sqrt{x}+2)(\sqrt{x}-2))`
`=(x+2\sqrt{x})/((\sqrt{x}+2)(\sqrt{x}-2))`
`=(\sqrt{x})\sqrt{x}+2))/((\sqrt{x}+2)(\sqrt{x}-2))`
`=\sqrt{x}/(\sqrt{x}-2)`
`2)C=\sqrt{x}/(\sqrt{x}-1)+3/(\sqrt{x}+1)-(6\sqrt{x}-4)/(x-1)(x>=0,x\ne1)`
`=\sqrt{x}/(\sqrt{x}-1)+3/(\sqrt{x}+1)-(6\sqrt{x}-4)/((\sqrt{x}+1)(\sqrt{x}-1))`
`=(\sqrt{x}(\sqrt{x}+1)+3(\sqrt{x}-1)-6\sqrt{x}+4)/((\sqrt{x}+1)(\sqrt{x}-1))`
`=(x+\sqrt{x}+3\sqrt{x}-3-6\sqrt{x}+4)/((\sqrt{x}+1)(\sqrt{x}-1))`
`=(x-2\sqrt{x}+1)/((\sqrt{x}+1)(\sqrt{x}-1))`
`=(\sqrt{x}-1)^2/((\sqrt{x}+1)(\sqrt{x}-1))`
`=(\sqrt{x}-1)/(\sqrt{x}+1)`
`3)(1/(\sqrt{x}+3)+(\sqrt{x}+9)/(x-9))*\sqrt{x}/2(x>=9,x\ne9)`
`=(1/(\sqrt{x}+3)+(\sqrt{x}+9)/((\sqrt{x}+3)(\sqrt{x}-3)))*\sqrt{x}/2`
`=(\sqrt{x}-3+\sqrt{x}+9)/((\sqrt{x}+3)(\sqrt{x}-3))*\sqrt{x}/2`
`=(2\sqrt{x}+6)/((\sqrt{x}+3)(\sqrt{x}-3))*\sqrt{x}/2`
`=(2(\sqrt{x}+3))/((\sqrt{x}+3)(\sqrt{x}-3))*\sqrt{x}/2`
`=2/(\sqrt{x}-3)*\sqrt{x}/2`
`=\sqrt{x}/(\sqrt{x}-3)`
1:ĐKXĐ: x>=0; x<>4
Ta có: \(A=\frac{x}{x-4}+\frac{1}{\sqrt{x}-2}+\frac{1}{\sqrt{x}+2}\)
\(=\frac{x}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}+\frac{\sqrt{x}+2}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}+\frac{\sqrt{x}-2}{\left(\sqrt{x}+2\right)\left(\sqrt{x}-2\right)}\)
\(=\frac{x+\sqrt{x}+2+\sqrt{x}-2}{\left(\sqrt{x}+2\right)\left(\sqrt{x}-2\right)}=\frac{x+2\sqrt{x}}{\left(\sqrt{x}+2\right)\left(\sqrt{x}-2\right)}=\frac{\sqrt{x}\left(\sqrt{x}+2\right)}{\left(\sqrt{x}+2\right)\left(\sqrt{x}-2\right)}=\frac{\sqrt{x}}{\sqrt[2]{x}-2}\)
2:
ĐKXĐ: x>=0; x<>1
Ta có: \(C=\frac{\sqrt{x}}{\sqrt{x}-1}+\frac{3}{\sqrt{x}+1}-\frac{6\sqrt{x}-4}{x-1}\)
\(=\frac{\sqrt{x}}{\sqrt{x}-1}+\frac{3}{\sqrt{x}+1}-\frac{6\sqrt{x}-4}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\)
\(=\frac{\sqrt{x}\left(\sqrt{x}+1\right)+3\left(\sqrt{x}-1\right)-6\sqrt{x}+4}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\)
\(=\frac{x+\sqrt{x}+3\sqrt{x}-3-6\sqrt{x}+4}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}=\frac{x-2\sqrt{x}+1}{\left(\sqrt{x}-1\right)\cdot\left(\sqrt{x}+1\right)}\)
\(=\frac{\left(\sqrt{x}-1\right)^2}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}=\frac{\sqrt{x}-1}{\sqrt{x}+1}\)
3:
ĐKXĐ: x>=0; x<>9
Ta có: \(M=\left(\frac{1}{\sqrt{x}+3}+\frac{\sqrt{x}+9}{x-9}\right)\cdot\frac{\sqrt{x}}{2}\)
\(=\left(\frac{1}{\sqrt{x}+3}+\frac{\sqrt{x}+9}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-3\right)}\right)\cdot\frac{\sqrt{x}}{2}\)
\(=\frac{\sqrt{x}-3+\sqrt{x}+9}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-3\right)}\cdot\frac{\sqrt{x}}{2}=\frac{2\sqrt{x}+6}{2\left(\sqrt{x}+3\right)}\cdot\frac{\sqrt{x}}{\sqrt{x}-3}=\frac{\sqrt{x}}{\sqrt{x}-3}\)
4:
ĐKXĐ: x>0; x<>9
Ta có: \(B=\left(\frac{\sqrt{x}+3}{x-9}+\frac{1}{\sqrt{x}+3}\right):\frac{\sqrt{x}}{\sqrt{x}-3}\)
\(=\left(\frac{\sqrt{x}+3}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-3\right)}+\frac{1}{\sqrt{x}+3}\right)\cdot\frac{\sqrt{x}-3}{\sqrt{x}}\)
\(=\frac{\sqrt{x}+3+\sqrt{x}-3}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-3\right)}\cdot\frac{\sqrt{x}-3}{\sqrt{x}}=\frac{2\sqrt{x}}{\sqrt{x}}\cdot\frac{1}{\sqrt{x}+3}=\frac{2}{\sqrt{x}+3}\)
5:
ĐKXĐ: x>=0; x<>9
ta có: \(P=\left(\frac{2\sqrt{x}}{\sqrt{x}+3}+\frac{\sqrt{x}}{\sqrt{x}-3}\right):\frac{\sqrt{x}+1}{\sqrt{x}-3}\)
\(=\frac{2\sqrt{x}\left(\sqrt{x}-3\right)+\sqrt{x}\left(\sqrt{x}+3\right)}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-3\right)}\cdot\frac{\sqrt{x}-3}{\sqrt{x}+1}\)
\(=\frac{2x-6\sqrt{x}+x+3\sqrt{x}}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-3\right)}\cdot\frac{\sqrt{x}-3}{\sqrt{x}+1}=\frac{3x-3\sqrt{x}}{\sqrt{x}+3}\cdot\frac{1}{\sqrt{x}+1}=\frac{3\sqrt{x}\left(\sqrt{x}-1\right)}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-1\right)}\)
6:
ĐKXĐ: x>=0; x<>9
Ta có: \(M=\frac{2\sqrt{x}}{\sqrt{x}+3}+\frac{\sqrt{x}+1}{\sqrt{x}-3}+\frac{11\sqrt{x}-3}{x-9}\)
\(=\frac{2\sqrt{x}}{\sqrt{x}+3}+\frac{\sqrt{x}+1}{\sqrt{x}-3}+\frac{11\sqrt{x}-3}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-3\right)}\)
\(=\frac{2\sqrt{x}\left(\sqrt{x}-3\right)+\left(\sqrt{x}+1\right)\left(\sqrt{x}+3\right)+11\sqrt{x}-3}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-3\right)}\)
\(=\frac{2x-6\sqrt{x}+x+4\sqrt{x}+3+11\sqrt{x}-3}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-1\right)}=\frac{3x+9\sqrt{x}}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-1\right)}\)
\(=\frac{3\sqrt{x}\left(\sqrt{x}+3\right)}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-1\right)}=\frac{3\sqrt{x}}{\sqrt{x}-1}\)
7:
ĐKXĐ: x>0; x<>1
Ta có: \(P=\left(\frac{\sqrt{x}}{\sqrt{x}-1}+\frac{\sqrt{x}}{x-1}\right):\left(\frac{2}{x}-\frac{2-x}{x\left(\sqrt{x}+1\right)}\right)\)
\(=\frac{\sqrt{x}\left(\sqrt{x}+1\right)+\sqrt{x}}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}:\frac{2\left(\sqrt{x}+1\right)-2+x}{x\left(\sqrt{x}+1\right)}\)
\(=\frac{x+\sqrt{x}+\sqrt{x}}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\cdot\frac{x\left(\sqrt{x}+1\right)}{2\sqrt{x}+2-2+x}\)
\(=\frac{x+2\sqrt{x}}{\sqrt{x}-1}\cdot\frac{x}{x+2\sqrt{x}}=\frac{x}{\sqrt{x}-1}\)
8:
ĐKXĐ: x>0; x<>9
Ta có: \(P=\left(\frac{\sqrt{x}}{x-9}-\frac{1}{\sqrt{x}+3}\right):\left(\frac{\sqrt{x}}{\sqrt{x}-3}-\frac{1}{\sqrt{x}}\right)\)
\(=\left(\frac{\sqrt{x}}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+3\right)}-\frac{1}{\sqrt{x}+3}\right):\frac{x-\left(\sqrt{x}-3\right)}{\sqrt{x}\left(\sqrt{x}-3\right)}\)
\(=\frac{\sqrt{x}-\left(\sqrt{x}-3\right)}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+3\right)}:\frac{x-\sqrt{x}+3}{\sqrt{x}\left(\sqrt{x}-3\right)}=\frac{3}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+3\right)}\cdot\frac{\sqrt{x}\left(\sqrt{x}-3\right)}{x-\sqrt{x}+3}\)
\(=\frac{3\sqrt{x}}{\left(\sqrt{x}+3\right)\left(x-\sqrt{x}+3\right)}\)
9:
ĐKXĐ: x>=0; x<>4
Ta có: \(M=\frac{x+12}{x-4}+\frac{1}{\sqrt{x}+2}-\frac{4}{\sqrt{x}-2}\)
\(=\frac{x+12}{\left(\sqrt{x}+2\right)\left(\sqrt{x}-2\right)}^{}+\frac{\sqrt{x}-2}{\left(\sqrt{x}+2\right)\left(\sqrt{x}-2\right)}-\frac{4\left(\sqrt{x}+2\right)}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}\)
\(=\frac{x+12+\sqrt{x}-2-4\sqrt{x}-8}{\left(\sqrt{x}+2\right)\left(\sqrt{x}-2\right)}=\frac{x-3\sqrt{x}+2}{\left(\sqrt{x}+2\right)\left(\sqrt{x}-2\right)}\)
\(=\frac{\left(\sqrt{x}-2\right)\left(\sqrt{x}-1\right)}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}=\frac{\sqrt{x}-1}{\sqrt{x}+2}\)
10:
ĐKXĐ: x>=0; x<>1
Ta có: \(P=\left(\frac{1}{\sqrt{x}-1}+\frac{\sqrt{x}}{x-1}\right):\left(\frac{\sqrt{x}}{\sqrt{x}-1}-1\right)\)
\(=\left(\frac{1}{\sqrt{x}-1}+\frac{\sqrt{x}}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\right):\frac{\sqrt{x}-\left(\sqrt{x}-1\right)}{\sqrt{x}-1}\)
\(=\frac{\sqrt{x}+1+\sqrt{x}}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}:\frac{1}{\sqrt{x}-1}=\frac{2\sqrt{x}+1}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\cdot\left(\sqrt{x}-1\right)=\frac{2\sqrt{x}+1}{\sqrt{x}+1}\)
11:
ĐKXĐ: x>=0; x<>25
Ta có: \(M=\left(\frac{x+3\sqrt{x}}{x-25}+\frac{1}{\sqrt{x}+5}\right):\frac{\sqrt{x}+2}{\sqrt{x}-5}\)
\(=\left(\frac{x+3\sqrt{x}}{\left(\sqrt{x}-5\right)\left(\sqrt{x}+5\right)}+\frac{1}{\sqrt{x}+5}\right):\frac{\sqrt{x}+2}{\sqrt{x}-5}\)
\(=\left(\frac{x+3\sqrt{x}}{\left(\sqrt{x}-5\right)\left(\sqrt{x}+5\right)}+\frac{\sqrt{x}-5}{\left(\sqrt{x}+5\right)\left(\sqrt{x}-5\right)}\right):\frac{\sqrt{x}+2}{\sqrt{x}-5}\)
\(=\frac{x+3\sqrt{x}+\sqrt{x}-5}{\left(\sqrt{x}-5\right)\left(\sqrt{x}+5\right)}:\frac{\sqrt{x}+2}{\sqrt{x}-5}\)
\(=\frac{x+4\sqrt{x}-5}{\left(\sqrt{x}+5\right)\left(\sqrt{x}-5\right)}\cdot\frac{\sqrt{x}-5}{\sqrt{x}+2}=\frac{\left(\sqrt{x}+5\right)\left(\sqrt{x}-1\right)}{\sqrt{x}+5}\cdot\frac{1}{\sqrt{x}+2}=\frac{\sqrt{x}-1}{\sqrt{x}+2}\)
12:
ĐKXĐ: x>=0; x<>9
Ta có: \(M=\left(\frac{x+\sqrt{x}-10}{x-9}-\frac{1}{\sqrt{x}-3}\right):\frac{1}{\sqrt{x}-3}\)
\(=\left(\frac{x+\sqrt{x}-10}{\left(\sqrt{x}-3\right)\cdot\left(\sqrt{x}+3\right)}-\frac{1}{\sqrt{x}-3}\right):\frac{1}{\sqrt{x}-3}\)
\(=\left(\frac{x+\sqrt{x}-10}{\left(\sqrt{x}-3\right)\cdot\left(\sqrt{x}+3\right)}-\frac{\sqrt{x}+3}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+3\right)}\right):\frac{1}{\sqrt{x}-3}\)
\(=\frac{x+\sqrt{x}-10-\sqrt{x}-3}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+3\right)}\cdot\left(\sqrt{x}-3\right)=\frac{x-13}{\sqrt{x}+3}\)
13:
ĐKXĐ: x>=0; x<>1
ta có: \(M=\left(\frac{1}{\sqrt{x}-1}+\frac{1}{\sqrt{x}+1}\right)\cdot\frac{\sqrt{x}-1}{2}\)
\(=\frac{\sqrt{x}+1+\sqrt{x}-1}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}\cdot\frac{\sqrt{x}-1}{2}\)
\(=\frac{2\sqrt{x}}{2\left(\sqrt{x}+1\right)}=\frac{\sqrt{x}}{\sqrt{x}+1}\)
14:
ĐKXĐ: x>=0; x<>4
ta có: \(M=\left(\frac{x}{x-4}+\frac{1}{\sqrt{x}-2}\right):\frac{2}{\sqrt{x}-2}\)
\(=\left(\frac{x}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}+\frac{1}{\sqrt{x}-2}\right):\frac{2}{\sqrt{x}-2}\)
\(=\frac{x+\sqrt{x}+2}{\left(\sqrt{x}+2\right)\left(\sqrt{x}-2\right)}\cdot\frac{\sqrt{x}-2}{2}=\frac{x+\sqrt{x}+2}{2\left(\sqrt{x}+2\right)}\)
15:
ĐKXĐ: x>0; x∉{4;1}
ta có: \(P=\left(1-\frac{4}{\sqrt{x}+1}+\frac{1}{\sqrt{x}-1}\right):\frac{x-2\sqrt{x}}{x-1}\)
\(=\frac{x-1-4\left(\sqrt{x}-1\right)+\sqrt{x}+1}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}\cdot\frac{x-1}{x-2\sqrt{x}}\)
\(=\frac{x-1-4\sqrt{x}+4+\sqrt{x}+1}{x-1}\cdot\frac{x-1}{\sqrt{x}\left(\sqrt{x}-2\right)}=\frac{x-3\sqrt{x}+4}{\sqrt{x}\left(\sqrt{x}-1\right)}\)
