rút gọn $\frac{1}{\sqrt[2]{2}+\sqrt[2]{2+\sqrt[2]{3}}}$+ $\frac{1}{\sqrt[2]{2}-\sqrt[2]{2-\sqrt[2]{3}}}$

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1. Đáp án:

   $- \dfrac{2\sqrt6}{3}$ 

   Giải thích các bước giải:

   $\begin{array}{l}\dfrac{1}{\sqrt2 + \sqrt{2 + \sqrt3}}+\dfrac{1}{\sqrt2 + \sqrt{2 -\sqrt3}}\\ = \dfrac{\sqrt2}{2 + \sqrt{4 + 2\sqrt3}}+\dfrac{\sqrt2}{2 + \sqrt{4 – 2\sqrt3}}\\ = \dfrac{\sqrt2}{2 + \sqrt3 + 1}+ \dfrac{\sqrt2}{2 + \sqrt3 – 1}\\ = \dfrac{\sqrt2}{3 + \sqrt3} + \dfrac{\sqrt2}{1 – \sqrt3}\\ = \dfrac{\sqrt2.(1 – \sqrt3)}{\sqrt3(1 + \sqrt3)(1 – \sqrt3)} + \dfrac{\sqrt2.\sqrt3(1 + \sqrt3)}{\sqrt3(1 + \sqrt3)(1 – \sqrt3)}\\ = \dfrac{\sqrt2 – \sqrt6 + \sqrt6 + 3\sqrt2}{\sqrt3(1 + \sqrt3)(1 – \sqrt3)}\\ = \dfrac{4\sqrt2}{\sqrt3.(1 – 3)}\\ = – \dfrac{2\sqrt6}{3}\end{array}$

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2. `1/(sqrt{2}+sqrt{(2+sqrt{3})})+1/(sqrt{2}-sqrt{(2-sqrt{3})})`

   `=(sqrt{2})/(2+sqrt{4+2sqrt{3}})+(sqrt{2})/(2-sqrt{4-2sqrt{3}})`

   `=(sqrt{2})/(2+sqrt{3}+1)+(sqrt{2})/(2-sqrt{3}-1)`

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

   `=(sqrt{2})/[sqrt{3}(sqrt{3}+1)]+(sqrt{2})/[sqrt{3}(sqrt{3}-1)]`

   `=(sqrt{6}-sqrt{2}+sqrt{6}+sqrt{2})/(2sqrt{3})`

   `=sqrt{2}`

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