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Lời giải:
Xét tử thức:
\(\frac{x-\sqrt{x}}{\sqrt{x}-1}-\frac{\sqrt{x}+1}{x+\sqrt{x}}=\frac{\sqrt{x}(\sqrt{x}-1)}{\sqrt{x}-1}-\frac{\sqrt{x}+1}{\sqrt{x}(\sqrt{x}+1)}=\sqrt{x}-\frac{1}{\sqrt{x}}=\frac{x-1}{\sqrt{x}}\)
\(\Rightarrow C=\frac{x-1}{\sqrt{x}}: \frac{\sqrt{x}+1}{x}=\frac{(\sqrt{x}-1)(\sqrt{x}+1)}{\sqrt{x}}.\frac{x}{\sqrt{x}+1}=\sqrt{x}(\sqrt{x}-1)\)
\( \left( \dfrac{ 1 }{ \sqrt{ x \phantom{\tiny{!}}} -1 } - \dfrac{ 1 }{ \sqrt{ x \phantom{\tiny{!}}} } \right) \left( \dfrac{ \sqrt{ x \phantom{\tiny{!}}} +1 }{ \sqrt{ x \phantom{\tiny{!}}} -2 } - \dfrac{ \sqrt{ x \phantom{\tiny{!}}} +2 }{ \sqrt{ x \phantom{\tiny{!}}} -1 } \right) \)
\(=\left(\dfrac{\sqrt{x}}{\sqrt{x}\left(\sqrt{x}-1\right)}-\dfrac{\sqrt{x}-1}{\sqrt{x}\left(\sqrt{x}-1\right)}\right)\left(\dfrac{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}{\left(\sqrt{x}-2\right)\left(\sqrt{x}-1\right)}-\dfrac{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}{\left(\sqrt{x}-2\right)\left(\sqrt{x}-1\right)}\right)\)
\(=\dfrac{\sqrt{x}-\sqrt{x}+1}{\sqrt{x}\left(\sqrt{x}-1\right)}\left(\dfrac{x-1}{\left(\sqrt{x}-2\right)\left(\sqrt{x}-1\right)}-\dfrac{x-4}{\left(\sqrt{x}-2\right)\left(\sqrt{x}-1\right)}\right)\)
\(=\dfrac{1}{\sqrt{x}\left(\sqrt{x}-1\right)}.\dfrac{3}{\left(\sqrt{x}-2\right)\left(\sqrt{x}-1\right)}\)
\(=\dfrac{3}{\sqrt{x}\left(\sqrt{x}-2\right)\left(\sqrt{x}-1\right)^2}\)
Ta có: \(\sqrt{48-2\sqrt{135}}-\sqrt{45}+\sqrt{18}\)
\(=\sqrt{45-2\cdot\sqrt{45}\cdot\sqrt{3}+3}-\sqrt{45}+\sqrt{18}\)
\(=\sqrt{\left(\sqrt{45}-\sqrt{3}\right)^2}-\sqrt{45}+\sqrt{18}\)
\(=\sqrt{45}-\sqrt{3}-\sqrt{45}+\sqrt{18}\)
\(=\sqrt{18}-\sqrt{3}\)
\( \dfrac{ 1 }{ \sqrt{ x \phantom{\tiny{!}}} -3 } + \dfrac{ 4 }{ \sqrt{ x \phantom{\tiny{!}}} +3 } - \dfrac{ 9- \sqrt{ x \phantom{\tiny{!}}} }{ x-9 } \)(ĐKXĐ: \(\left\{{}\begin{matrix}x>=0\\x< >9\end{matrix}\right.\))
\(=\dfrac{1}{\sqrt{x}-3}+\dfrac{4}{\sqrt{x}+3}+\dfrac{\sqrt{x}-9}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+3\right)}\)
\(=\dfrac{\sqrt{x}+3+4\left(\sqrt{x}-3\right)+\sqrt{x}-9}{\left(\sqrt{x}-3\right)\cdot\left(\sqrt{x}+3\right)}\)
\(=\dfrac{2\sqrt{x}-6+4\sqrt{x}-12}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+3\right)}=\dfrac{6\sqrt{x}-18}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+3\right)}\)
\(=6\cdot\dfrac{\left(\sqrt{x}-3\right)}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+3\right)}=\dfrac{6}{\sqrt{x}+3}\)
ĐKXĐ:...
\(A=\left(\frac{\sqrt{a}+2}{\sqrt{a}\left(\sqrt{a}+2\right)}-\frac{\sqrt{a}-1}{\left(\sqrt{a}-1\right)\left(\sqrt{a}+1\right)}\right).\frac{\sqrt{a}+1}{\sqrt{a}}=\left(\frac{1}{\sqrt{a}}-\frac{1}{\sqrt{a}+1}\right).\frac{\left(\sqrt{a}+1\right)}{\sqrt{a}}\)
\(=\frac{1}{\sqrt{a}\left(\sqrt{a}+1\right)}.\frac{\left(\sqrt{a}+1\right)}{\sqrt{a}}=\frac{1}{a}\)
\(C=\left(\frac{\left(\sqrt{x}+1\right)\left(2\sqrt{x}-1\right)}{-\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}+\frac{\sqrt{x}\left(\sqrt{x}+1\right)\left(2\sqrt{x}-1\right)}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}\right).\frac{\sqrt{x}\left(\sqrt{x}-1\right)}{2\sqrt{x}-1}\)
\(=\left(\frac{\left(\sqrt{x}+1\right)}{-\left(\sqrt{x}+1\right)}+\frac{\sqrt{x}\left(\sqrt{x}+1\right)}{x+\sqrt{x}+1}\right).\frac{\sqrt{x}\left(\sqrt{x}-1\right)}{\left(2\sqrt{x}-1\right)}.\frac{\left(2\sqrt{x}-1\right)}{\left(\sqrt{x}-1\right)}\)
\(=\left(-1+\frac{\sqrt{x}\left(\sqrt{x}+1\right)}{x+\sqrt{x}+1}\right).\sqrt{x}=\left(\frac{-x-\sqrt{x}-1+x+\sqrt{x}}{x+\sqrt{x}+1}\right)\sqrt{x}=\frac{-\sqrt{x}}{x+\sqrt{x}+1}\)