Đáp án + Giải thích các bước giải:

`1)1/{2-\sqrt{3}} + 1/{2+\sqrt{3}}`

`={2+\sqrt{3} + 2 - \sqrt{3}}/ {(2+\sqrt{3})(2-\sqrt{3})}`

`=4/{4-3}=4.`

`2)1/{2-\sqrt{5}} - 1/{2+\sqrt{5}}`

`={2+\sqrt{5}}/{(2+\sqrt{5})(2-\sqrt{5})}- {2-\sqrt{5}}/{(2+\sqrt{5})(2-\sqrt{5})}`

`={2+\sqrt{5}-2+\sqrt{5})/{(2+\sqrt{5})(2-\sqrt{5})}`

`={2\sqrt{5}}/{4-5}=-2\sqrt{5}.`

`3)` ĐKXĐ: `x\ne0,x\ne1.`

`{x-1}/{\sqrt{x}-1}+{x-1}/{\sqrt{x}+1}`

`={(\sqrt{x}-1)(\sqrt{x}+1)}/{\sqrt{x}-1}+{(\sqrt{x}-1)(\sqrt{x}+1)}/{\sqrt{x}+1}`

`=\sqrt{x}+1 + \sqrt{x}-1`

`=2\sqrt{x}.`

`4)`ĐKXĐ: `x\ne1,x\ne4.`

`{\sqrt{x}-1}/{x-1}-{\sqrt{x}+2}/{x-4}`

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

`=1/{\sqrt{x}+1} - 1/{\sqrt{x}-2}`

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

`=-3/{x-\sqrt{x}-2}.`

`5)`  ĐKXĐ: `x\ne0,x\ne1.`

`{1+x\sqrt{x}}/{1+\sqrt{x}} + {1-x\sqrt{x}}/{1-\sqrt{x}}`

`={1+(\sqrt{x})^3}/{1+\sqrt{x}} + {1-(\sqrt{x})^3}/{1-\sqrt{x}}`

`={(1+\sqrt{x})(1-\sqrt{x}+x)}/{1+\sqrt{x}} + {(1-\sqrt{x})(1+\sqrt{x}+x)}/{1-\sqrt{x}}`

`=1-\sqrt{x}+x+ 1+\sqrt{x}+x`

`=2+2x.`

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Chú ý: `x\sqrt{x}=\sqrt{x}.\sqrt{x}.\sqrt{x}=(\sqrt{x})^3.`

Đáp án:

Giải thích các bước giải:

 1)$\dfrac{1}{2-\sqrt{3}}+\dfrac{1}{2+\sqrt{3}}$

$=2+\sqrt{3}+2-\sqrt{3}$

$=4$

2)$\dfrac{1}{2-\sqrt{5}}-\dfrac{1}{2+\sqrt{5}}$

$=-(2+\sqrt{5})+2-\sqrt{5}$

$=-2\sqrt{5}$

3)$\dfrac{x-1}{\sqrt{x}-1}+\dfrac{x-1}{\sqrt{x}+1}$

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

$=2\sqrt{x}$

4)$\dfrac{\sqrt{x}-1}{x-1}-\dfrac{\sqrt{x}+2}{x-4}$

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

$=\dfrac{1}{\sqrt{x}+1}-\dfrac{1}{\sqrt{x}-2}$

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

$=\dfrac{-3}{(\sqrt{x}+1).(\sqrt{x}-2)}$

8)$\dfrac{3\sqrt{x}+x}{\sqrt{x}+3}-(\sqrt{x}+1)$

$=\sqrt{x}-\sqrt{x}-1$

$=-1$