Question

\(\left(\dfrac{2\sqrt{x}-x}{x\sqrt{x-1}}-\dfrac{1}{\sqrt{x}-1}\right)\div1\left(-\dfrac{\sqrt{x}+2}{x+\sqrt{x}+1}\right)\left(x>0,x\ne1\right)\)

Answer 1

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

\(=\left[\dfrac{\sqrt{x}\left(2-\sqrt{x}\right)}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}-\dfrac{1}{\sqrt{x}-1}\right]\div\left[\dfrac{\left(x+\sqrt{x}+1\right)-\left(\sqrt{x}+2\right)}{x+\sqrt{x}+1}\right]\)

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

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