Câu 1:

Sửa đề: \(B=\left(\dfrac{x}{x+3\sqrt{x}}+\dfrac{1}{\sqrt{x}+3}\right):\left(1-\dfrac{2}{\sqrt{x}}+\dfrac{6}{x+3\sqrt{x}}\right)\)

Ta có: \(B=\left(\dfrac{x}{x+3\sqrt{x}}+\dfrac{1}{\sqrt{x}+3}\right):\left(1-\dfrac{2}{\sqrt{x}}+\dfrac{6}{x+3\sqrt{x}}\right)\)

\(=\left(\dfrac{x}{\sqrt{x}\left(\sqrt{x}+3\right)}+\dfrac{1}{\sqrt{x}+3}\right):\left(\dfrac{x+3\sqrt{x}-2\left(\sqrt{x}+3\right)+6}{\sqrt{x}\left(\sqrt{x}+3\right)}\right)\)

\(=\dfrac{\sqrt{x}+1}{\sqrt{x}+3}:\dfrac{x+3\sqrt{x}-2\sqrt{x}-6+6}{\sqrt{x}\left(\sqrt{x}+3\right)}\)

\(=\dfrac{\sqrt{x}+1}{\sqrt{x}+3}\cdot\dfrac{\sqrt{x}\left(\sqrt{x}+3\right)}{x+\sqrt{x}}\)

\(=\dfrac{\sqrt{x}\left(\sqrt{x}+1\right)}{\sqrt{x}\left(\sqrt{x}+1\right)}=1\)

Câu 3: 

Ta có: \(Q=\left(\dfrac{a}{a-2\sqrt{a}}+\dfrac{a}{\sqrt{a}-2}\right):\dfrac{\sqrt{a}+1}{a-4\sqrt{a}+4}\)

\(=\left(\dfrac{a}{\sqrt{a}\left(\sqrt{a}-2\right)}+\dfrac{a}{\sqrt{a}-2}\right):\dfrac{\sqrt{a}+1}{\left(\sqrt{a}-2\right)^2}\)

\(=\dfrac{a+\sqrt{a}}{\sqrt{a}-2}\cdot\dfrac{\sqrt{a}-2}{\sqrt{a}+1}\cdot\dfrac{\sqrt{a}-2}{1}\)

\(=\sqrt{a}\left(\sqrt{a}-2\right)\)

\(=a-2\sqrt{a}\)

\(\frac{2x-1+\sqrt{x}}{1-x}+\frac{2x\cdot\sqrt{x}+x-\sqrt{x}}{1+x\cdot\sqrt{x}}\)

\(=\frac{2x+\sqrt{x}-1}{1-x}+\frac{\sqrt{x}\left(2x+\sqrt{x}-1\right)}{\left(\sqrt{x}+1\right)\left(x-\sqrt{x}+1\right)}\)

\(=\left(2x+\sqrt{x}-1\right)\left\lbrack-\frac{1}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}+\frac{\sqrt{x}}{\left(x-\sqrt{x}+1\right)\left(\sqrt{x}+1\right)}\right\rbrack\)

\(=\left(\sqrt{x}+1\right)\left(2\sqrt{x}-1\right)\cdot\left\lbrack\frac{-x+\sqrt{x}-1+\sqrt{x}\left(\sqrt{x}-1\right)}{\left(x-\sqrt{x}+1\right)\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}\right\rbrack\)

\(=\left(\sqrt{x}+1\right)\left(2\sqrt{x}-1\right)\cdot\frac{-1}{\left(x-\sqrt{x}+1\right)\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}=\frac{-\left(2\sqrt{x}-1\right)}{\left(x-\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}\)

Ta có: \(1-\left(\frac{2x-1+\sqrt{x}}{1-x}+\frac{2x\cdot\sqrt{x}+x-\sqrt{x}}{1+x\cdot\sqrt{x}}\right)\cdot\frac{\left(x-\sqrt{x}\right)\left(1-\sqrt{x}\right)}{2\sqrt{x}-1}\)

\(=1+\frac{\left(2\sqrt{x}-1\right)}{\left(x-\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}\cdot\frac{\sqrt{x}\left(\sqrt{x}-1\right)\left(1-\sqrt{x}\right)}{2\sqrt{x}-1}\)

\(=1+\frac{\sqrt{x}\left(1-\sqrt{x}\right)}{x-\sqrt{x}+1}=\frac{x-\sqrt{x}+1+\sqrt{x}-x}{x-\sqrt{x}+1}=\frac{1}{x-\sqrt{x}+1}\)

Sửa đề: \(A=\left(\frac{2x-1+\sqrt{x}}{1-x}+\frac{2x\cdot\sqrt{x}+x-\sqrt{x}}{1+x\cdot\sqrt{x}}\right)\cdot\frac{\left(x-\sqrt{x}\right)\left(1-\sqrt{x}\right)}{2\sqrt{x}-1}-1\)

Ta có: \(\frac{2x-1+\sqrt{x}}{1-x}+\frac{2x\cdot\sqrt{x}+x-\sqrt{x}}{1+x\cdot\sqrt{x}}\)

\(=\left(2x+\sqrt{x}-1\right)\cdot\left(\frac{-1}{x-1}+\frac{\sqrt{x}}{\left(\sqrt{x}+1\right)\left(x-\sqrt{x}+1\right)}\right)\)

\(=\left(2x+2\sqrt{x}-\sqrt{x}-1\right)\cdot\left(\frac{-1}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}+\frac{\sqrt{x}}{\left(\sqrt{x}+1\right)\left(x-\sqrt{x}+1\right)}\right)\)

\(=\left(\sqrt{x}+1\right)\left(2\sqrt{x}-1\right)\cdot\frac{-x+\sqrt{x}-1+\sqrt{x}\left(\sqrt{x}-1\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)\left(x-\sqrt{x}+1\right)}\)

\(=\left(2\sqrt{x}-1\right)\cdot\frac{-1}{\left(\sqrt{x}-1\right)\left(x-\sqrt{x}+1\right)}=\frac{-\left(2\sqrt{x}-1\right)}{\left(\sqrt{x}-1\right)\left(x-\sqrt{x}+1\right)}\)

Ta có: \(A=\left(\frac{2x-1+\sqrt{x}}{1-x}+\frac{2x\cdot\sqrt{x}+x-\sqrt{x}}{1+x\cdot\sqrt{x}}\right)\cdot\frac{\left(x-\sqrt{x}\right)\left(1-\sqrt{x}\right)}{2\sqrt{x}-1}-1\)

\(=\frac{-\left(2\sqrt{x}-1\right)}{\left(\sqrt{x}-1\right)\left(x-\sqrt{x}+1\right)}\cdot\frac{-\left(x-\sqrt{x}\right)\left(\sqrt{x}-1\right)}{2\sqrt{x}-1}-1\)

\(=\frac{x-\sqrt{x}}{x-\sqrt{x}+1}-1=\frac{x-\sqrt{x}-x+\sqrt{x}-1}{x-\sqrt{x}+1}=\frac{-1}{x-\sqrt{x}+1}\)

???

\(B=\left(\dfrac{2x+1}{x\sqrt{x}-1}-\dfrac{\sqrt{x}}{x+\sqrt{x}+1}\right)\left(\dfrac{1+x\sqrt{x}}{1+\sqrt{x}}-\sqrt{x}\right)+\dfrac{2-2\sqrt{x}}{\sqrt{x}}(x \geq 0,x \neq 1\)
`=((2x+1-x+\sqrtx)/(x\sqrtx-1))(((\sqrtx+1)(x-\sqrtx+1))/(\sqrtx+1)-\sqrtx)+(2-2sqrtx)/sqrtx`
`=((x-\sqrtx+1)/((\sqrtx-1))(x+sqrtx+1)))(x-2\sqrtx+1)-(2\sqrtx-2)/sqrtx`
`=(1/(\sqrtx-1))(\sqrtx-1)^2-(2(\sqrtx-1))/sqrtx`
`=\sqrtx-1-(2(\sqrtx-1))/sqrtx`
`=(x-\sqrtx-2\sqrtx+2)/sqrtx`
`=(x-3sqrtx+2)/sqrtx`

a) \(\sqrt{36\left(x-5\right)^2}\left(x\ge5\right)=6\left|x-5\right|=6\left(x-5\right)=6x-30\)

b) \(\sqrt{\dfrac{1}{4}\left(1-x\right)^2}\left(x>1\right)=\dfrac{1}{2}\left|1-x\right|=\dfrac{1}{2}\left(x-1\right)=\dfrac{1}{2}x-\dfrac{1}{2}\)

c) \(\sqrt{x^2\left(2x-4\right)^2}\left(x\ge2\right)=\left|x\left(2x-4\right)\right|=x\left(2x-4\right)=2x^2-4x\)

d) \(\dfrac{1}{x}\sqrt{x^2\left(1+x\right)^2}\left(x< -1\right)=\dfrac{1}{x}\left|x\left(1+x\right)\right|=\dfrac{1}{x}x\left(1+x\right)=1+x\)

1: \(\frac{x\sqrt{x}+x+\sqrt{x}}{x\sqrt{x}-1}-\frac{\sqrt{x}+3}{1-\sqrt{x}}\)

\(=\frac{\sqrt{x}\left(x+\sqrt{x}+1\right)}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}+\frac{\sqrt{x}+3}{\sqrt{x}-1}\)

\(=\frac{\sqrt{x}}{\sqrt{x}-1}+\frac{\sqrt{x}+3}{\sqrt{x}-1}=\frac{2\sqrt{x}+3}{\sqrt{x}-1}\)

\(\frac{x-1}{2x+\sqrt{x}-1}\)

\(=\frac{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}{2x+2\sqrt{x}-\sqrt{x}-1}\)

\(=\frac{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}{\left(\sqrt{x}+1\right)\left(2\sqrt{x}-1\right)}=\frac{\sqrt{x}-1}{2\sqrt{x}-1}\)

Ta có: \(B=\left(\frac{x\sqrt{x}+x+\sqrt{x}}{x\sqrt{x}-1}-\frac{\sqrt{x}+3}{1-\sqrt{x}}\right)\cdot\frac{x-1}{2x+\sqrt{x}-1}\)

\(=\frac{2\sqrt{x}+3}{\sqrt{x}-1}\cdot\frac{\sqrt{x}-1}{2\sqrt{x}-1}=\frac{2\sqrt{x}+3}{2\sqrt{x}-1}\)

2: \(\frac{\left(\sqrt{x}-1\right)^2\left(\sqrt{x}+1\right)^2}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\)

\(=\frac{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)\cdot\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}=\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)=x-1\)

Ta có: \(A=\frac{\left(\sqrt{x}-1\right)^2\left(\sqrt{x}+1\right)^2}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}-\frac{3\sqrt{x}+1}{x-1}\)

\(=x-1-\frac{3\sqrt{x}+1}{x-1}=\frac{\left(x-1\right)^2-3\sqrt{x}-1}{x-1}=\frac{x^2-2x-3\sqrt{x}}{x-1}\)

\(=\frac{\sqrt{x}\left(x\sqrt{x}-2\sqrt{x}-3\right)}{x-1}\)

ĐKXĐ: x>=0; x<>1

Ta có: \(\frac{2x-1}{x\sqrt{x}-1}-\frac{1}{\sqrt{x}-1}\)

\(=\frac{2x-1}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}-\frac{1}{\sqrt{x}-1}\)

\(=\frac{2x-1-x-\sqrt{x}-1}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}=\frac{x-\sqrt{x}-2}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}\)

\(=\frac{\left(\sqrt{x}-2\right)\left(\sqrt{x}+1\right)}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}\)

Ta có: \(1-\frac{x-2}{x+\sqrt{x}+1}\)

\(=\frac{x+\sqrt{x}+1-x+2}{x+\sqrt{x}+1}=\frac{\sqrt{x}+3}{x+\sqrt{x}+1}\)

Ta có: \(A=\left(\frac{2x-1}{x\sqrt{x}-1}-\frac{1}{\sqrt{x}-1}\right)\) :\(\left(1-\frac{x-2}{x+\sqrt{x}+1}\right)\)

\(=\frac{\left(\sqrt{x}-2\right)\left(\sqrt{x}+1\right)}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}:\frac{\sqrt{x}+3}{x+\sqrt{x}+1}=\frac{\left(\sqrt{x}-2\right)\left(\sqrt{x}+1\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+3\right)}\)

Ta có: \(\frac{1}{1-\sqrt{x}}-\frac{1}{\sqrt{x}}\)

\(=\frac{\sqrt{x}-1+\sqrt{x}}{\sqrt{x}\left(1-\sqrt{x}\right)}=\frac{2\sqrt{x}-1}{\sqrt{x}\left(1-\sqrt{x}\right)}\)

Ta có: \(\frac{2x+\sqrt{x}-1}{1-x}+\frac{2x\cdot\sqrt{x}+x-\sqrt{x}}{1+x\cdot\sqrt{x}}\)

\(=\left(2x+\sqrt{x}-1\right)\left(\frac{1}{1-x}+\frac{\sqrt{x}}{1+x\cdot\sqrt{x}}\right)\)

\(=\left(\sqrt{x}+1\right)\left(2\sqrt{x}-1\right)\cdot\left(\frac{-1}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}+\frac{\sqrt{x}}{\left(\sqrt{x}+1\right)\left(x-\sqrt{x}+1\right)}\right)\)

\(=\left(\sqrt{x}+1\right)\left(2\sqrt{x}-1\right)\cdot\frac{-x+\sqrt{x}-1+\sqrt{x}\left(\sqrt{x}-1\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)\left(x-\sqrt{x}+1\right)}\)

\(=\left(\sqrt{x}+1\right)\left(2\sqrt{x}-1\right)\cdot\frac{-1}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)\left(x-\sqrt{x}+1\right)}\)

\(=\frac{-\left(2\sqrt{x}-1\right)}{\left(\sqrt{x}-1\right)\left(x-\sqrt{x}+1\right)}\)

Ta có: \(\left(\frac{1}{1-\sqrt{x}}-\frac{1}{\sqrt{x}}\right):\left(\frac{2x+\sqrt{x}-1}{1-x}+\frac{2x\cdot\sqrt{x}+x-\sqrt{x}}{1+x\cdot\sqrt{x}}\right)\)

\(=\frac{2\sqrt{x}-1}{\sqrt{x}\left(1-\sqrt{x}\right)}:\frac{-\left(2\sqrt{x}-1\right)}{\left(\sqrt{x}-1\right)\left(x-\sqrt{x}+1\right)}\)

\(=\frac{-\left(2\sqrt{x}-1\right)}{\sqrt{x}\left(\sqrt{x}-1\right)}\cdot\frac{\left(\sqrt{x}-1\right)\left(x-\sqrt{x}+1\right)}{-\left(2\sqrt{x}-1\right)}=\frac{x-\sqrt{x}+1}{\sqrt{x}}\)

Ta có: \(B=\left(\dfrac{2x+1}{x\sqrt{x}+1}-\dfrac{1}{\sqrt{x}-1}\right):\left(1-\dfrac{x}{x+\sqrt{x}+1}\right)\)

\(=\dfrac{2x\sqrt{x}-2x+\sqrt{x}-1-x\sqrt{x}-1}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)\left(x-\sqrt{x}+1\right)}:\dfrac{x+\sqrt{x}+1-x}{x+\sqrt{x}+1}\)

\(=\dfrac{x\sqrt{x}-2x+\sqrt{x}-2}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)\left(x-\sqrt{x}+1\right)}\cdot\dfrac{x+\sqrt{x}+1}{\sqrt{x}+1}\)

\(=\dfrac{\left(x+1\right)\left(\sqrt{x}+1\right)\left(x+\sqrt{x}+1\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)^2\cdot\left(x-\sqrt{x}+1\right)}\)

\(=\dfrac{\left(x+1\right)\left(x+\sqrt{x}+1\right)}{\left(x-1\right)\left(x-\sqrt{x}+1\right)}\)

Ta có: \(B=\left(\dfrac{2x+1}{x\sqrt{x}+1}-\dfrac{1}{\sqrt{x}-1}\right):\left(1-\dfrac{x}{x+\sqrt{x}+1}\right)\)

\(=\dfrac{2x\sqrt{x}-2x+\sqrt{x}-1-x\sqrt{x}-1}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)\left(x-\sqrt{x}+1\right)}:\dfrac{x+\sqrt{x}+1-x}{x+\sqrt{x}+1}\)

\(=\dfrac{x\sqrt{x}-2x+\sqrt{x}-2}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)\left(x-\sqrt{x}+1\right)}\cdot\dfrac{x+\sqrt{x}+1}{\sqrt{x}+1}\)

\(=\dfrac{\left(\sqrt{x}-2\right)\left(x+1\right)\cdot\left(x+\sqrt{x}+1\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)^2\cdot\left(x-\sqrt{x}+1\right)}\)