\(a,ĐK:x>0;x\ne1\\ b,A=\dfrac{1+\sqrt{x}}{\sqrt{x}\left(\sqrt{x}-1\right)}\cdot\dfrac{\left(\sqrt{x}-1\right)^2}{\sqrt{x}+1}=\dfrac{\sqrt{x}-1}{\sqrt{x}}\\ c,x=4\Leftrightarrow\sqrt{x}=2\Leftrightarrow A=\dfrac{2-1}{2}=\dfrac{1}{2}\)

a: ĐKXĐ: \(x>0\)

b: Ta có: \(A=\dfrac{x^2+\sqrt{x}}{x-\sqrt{x}+1}-\dfrac{2x+\sqrt{x}}{\sqrt{x}}+1\)

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

\(=x-\sqrt{x}\)

`a)ĐKXĐ:{(x > 0),(x \ne 4):}`

`b)` Với `x > 0,x \ne 4` có:

`A=[\sqrt{x}(\sqrt{x}+2)+\sqrt{x}(\sqrt{x}-2)]/[x-4].[x-4]/[\sqrt{4x}]`

`A=[x-2\sqrt{x}+x-2\sqrt{x}]/[2\sqrt{x}]`

`A=[2\sqrt{x}(\sqrt{x}-2)]/[2\sqrt{x}]=\sqrt{x}-2`

`c)` Với `x > 0,x \ne 4` có:

`A < 3 <=>\sqrt{x}-2 < 3<=>\sqrt{x} < 5<=>x < 25`

           Kết hợp đk

 `=>0 < x < 25 ,x \ne 4`

a: ĐKXĐ: \(\left\{{}\begin{matrix}x\ge0\\x\ne1\end{matrix}\right.\)

Ta có: \(A=\dfrac{\sqrt{x}}{\sqrt{x}-1}+\dfrac{3}{\sqrt{x}+1}-\dfrac{6\sqrt{x}-4}{x-1}\)

\(=\dfrac{x+\sqrt{x}+3\sqrt{x}-3-6\sqrt{x}+4}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\)

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

Thay \(x=6-2\sqrt{5}\) vào A, ta được:

\(A=\dfrac{\sqrt{5}-1-1}{\sqrt{5}-1+1}=\dfrac{\sqrt{5}-2}{\sqrt{5}}=\dfrac{5-2\sqrt{5}}{5}\)

b: Để \(A< \dfrac{1}{2}\) thì \(\dfrac{\sqrt{x}-1}{\sqrt{x}+1}-\dfrac{1}{2}< 0\)

\(\Leftrightarrow2\sqrt{x}-2-\sqrt{x}-1< 0\)

\(\Leftrightarrow x< 9\)

Kết hợp ĐKXĐ, ta được: \(\left\{{}\begin{matrix}0\le x< 9\\x\ne1\end{matrix}\right.\)

a: ĐKXĐ: \(\left\{{}\begin{matrix}x>0\\x\ne4\end{matrix}\right.\)

b: Ta có: \(D=\left(\dfrac{\sqrt{x}+2}{\sqrt{x}-2}-\dfrac{\sqrt{x}}{\sqrt{x}+2}-\dfrac{5\sqrt{x}+5}{x-4}\right)\cdot\dfrac{x-4}{\sqrt{x}}\)

\(=\dfrac{x+4\sqrt{x}+4-x+4\sqrt{x}-5\sqrt{x}-5}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}\cdot\dfrac{x-4}{\sqrt{x}}\)

\(=\dfrac{3\sqrt{x}-1}{\sqrt{x}}\)

1) ĐKXĐ: \(x\notin\left\{0;1\right\}\)

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

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

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

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

a) ĐKXĐ: \(\left\{{}\begin{matrix}x>0\\x\ne4\end{matrix}\right.\)

Ta có: \(P=\left(\dfrac{\sqrt{x}}{\sqrt{x}-2}+\dfrac{\sqrt{x}}{\sqrt{x}+2}\right):\dfrac{2\sqrt{x}}{x-4}\)

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

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

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

\(=\sqrt{x}\)

b) Để P>4 thì \(\sqrt{x}>4\)

hay x>16

Kết hợp ĐKXĐ, ta được: x>16

Vậy: Khi x>16 thì P>4

\(a,dkxd:x\ge0,x\ne4\)

\(b,B=\left(\dfrac{\sqrt{x}}{\sqrt{x}-2}-\dfrac{4}{x-2\sqrt{x}}\right)\dfrac{1}{\sqrt{x}-2}\\ =\left(\dfrac{\sqrt{x}}{\sqrt{x}-2}-\dfrac{4}{\sqrt{x}\left(\sqrt{x}-2\right)}\right)\dfrac{1}{\sqrt{x}-2}\\ =\dfrac{\sqrt{x^2}-4}{\sqrt{x}\left(\sqrt{x}-2\right)}.\dfrac{1}{\sqrt{x}-2}\\ =\dfrac{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}{\sqrt{x}\left(\sqrt{x}-2\right)^2}\\ =\dfrac{\sqrt{x}+2}{\sqrt{x}\left(\sqrt{x}-2\right)}\)

\(c,x=16\left(tm\right)\Rightarrow B=\dfrac{\sqrt{16}+2}{\sqrt{16}\left(\sqrt{16}-2\right)}=\dfrac{4+2}{4\left(4-2\right)}=\dfrac{6}{8}=\dfrac{3}{4}\)

\(d,B>0\Leftrightarrow\dfrac{\sqrt{x}+2}{\sqrt{x}\left(\sqrt{x}-2\right)}>0\Leftrightarrow\sqrt{x}+2>0\Leftrightarrow\sqrt{x}>-2\left(ktm\right)\)

\(\Leftrightarrow\sqrt{x}\left(\sqrt{x}-2\right)< 0\Leftrightarrow\sqrt{x}< 2\Leftrightarrow x< 4\)

Kết hợp với \(dk:x\ge0\) ta kết luận \(0\le x< 4\) thì \(B>0\).

a) Điều kiện xác định:

\(\left\{{}\begin{matrix}x-2\sqrt{x}\ne0\\x\ge0\end{matrix}\right.\)\(\Leftrightarrow x>0,x\ne4\)

Vậy...

b) \(B=\dfrac{\sqrt{x}.\sqrt{x}-4}{\sqrt{x}\left(\sqrt{x}-2\right)}.\dfrac{1}{\sqrt{x}-2}\)

\(=\dfrac{x-4}{\sqrt{x}\left(\sqrt{x}-2\right)^2}=\dfrac{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}{\sqrt{x}\left(\sqrt{x}-2\right)^2}\)\(=\dfrac{\sqrt{x}+2}{\sqrt{x}\left(\sqrt{x}-2\right)}\)

Vậy \(B=\dfrac{\sqrt{x}+2}{\sqrt{x}\left(\sqrt{x}-2\right)}\)

c) Tại x=16 ( thỏa mãn đk) thay vào B đã rút gọn ta được:

\(B=\dfrac{\sqrt{16}+2}{\sqrt{16}\left(\sqrt{16}-2\right)}=\dfrac{3}{4}\)

d) \(B>0\Leftrightarrow\dfrac{\sqrt{x}+2}{\sqrt{x}\left(\sqrt{x}-2\right)}>0\)

\(\Leftrightarrow\sqrt{x}-2>0\)\(\Leftrightarrow\sqrt{x}>2\Leftrightarrow x>4\)

Vậy x>4 thì B>0

a: ĐKXĐ: \(\left\{{}\begin{matrix}x\ge0\\x\ne4\end{matrix}\right.\)

Ta có: \(P=\dfrac{\sqrt{x}}{\sqrt{x}+2}+\dfrac{2}{\sqrt{x}-2}-\dfrac{4\sqrt{x}}{x-4}\)

\(=\dfrac{x-2\sqrt{x}+2\sqrt{x}+4-4\sqrt{x}}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}\)

\(=\dfrac{\sqrt{x}-2}{\sqrt{x}+2}\)