Bài đầu : \(\dfrac{1}{\sqrt{x}+1}+\dfrac{x}{\sqrt{x}-x}\)

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

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

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

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

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

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

Bài cuối : \(\dfrac{x\sqrt{x}+1}{x-1}-\dfrac{x-1}{\sqrt{x}+1}\)

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

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

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

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

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

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

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