Ta có:

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

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

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

b.

\(x=\dfrac{1}{4}\) \(\Rightarrow A=-1-\dfrac{1}{\sqrt{\dfrac{1}{4}}}=-1-\dfrac{1}{\dfrac{1}{2}}=-3\)

c. Từ câu b ta có A= -3 khi x = 1/4

Ta có:

\(\dfrac{\sqrt{21}-\sqrt{7}}{\sqrt{3}-1}=\dfrac{\sqrt{7}\left(\sqrt{3}-1\right)}{\sqrt{3}-1}=\sqrt{7}\\ \dfrac{\sqrt{18}-\sqrt{3}}{\sqrt{6}-1}=\dfrac{\sqrt{3}\left(\sqrt{6}-1\right)}{\sqrt{6}-1}=\sqrt{3}\)

Vì vậy \(\left(\dfrac{\sqrt{21}-\sqrt{7}}{\sqrt{3}-1}+\dfrac{\sqrt{18}-\sqrt{3}}{\sqrt{6}-1}\right):\dfrac{2}{\sqrt{7}-\sqrt{3}}\\ =\left(\sqrt{7}+\sqrt{3}\right).\dfrac{\sqrt{7}-\sqrt{3}}{2}\\ =\dfrac{7-3}{2}=2\)

Đs...