\(=\dfrac{\sqrt{2}-\sqrt{2-\sqrt{3}}-\sqrt{2}-\sqrt{2+\sqrt{3}}}{\left(\sqrt{2}+\sqrt{2+\sqrt{3}}\right)\left(\sqrt{2}-\sqrt{2-\sqrt{3}}\right)}\\ =\dfrac{-\sqrt{2-\sqrt{3}}-\sqrt{2+\sqrt{3}}}{2-\sqrt{4-2\sqrt{3}}+\sqrt{4+2\sqrt{3}}-2+\sqrt{3}}\\ =\dfrac{-\sqrt{\left(\sqrt{\dfrac{3}{2}}-\sqrt{\dfrac{1}{2}}\right)^2}-\sqrt{\left(\sqrt{\dfrac{3}{2}}+\sqrt{\dfrac{1}{2}}\right)^2}}{\sqrt{3}-\sqrt{3}+1+\sqrt{3}+1}\\ =\dfrac{-\sqrt{\dfrac{3}{2}}+\sqrt{\dfrac{1}{2}}-\sqrt{\dfrac{3}{2}}-\sqrt{\dfrac{1}{2}}}{\sqrt{3}+2}\\ =\dfrac{-\sqrt{6}}{\sqrt{3}+2}=\dfrac{-\sqrt{6}\left(2-\sqrt{3}\right)}{1}=3\sqrt{2}-2\sqrt{6}\)
Lời giải:
\(P=\frac{\sqrt{2}-\sqrt{2+\sqrt{3}}}{2-(2+\sqrt{3})}-\frac{\sqrt{2}+\sqrt{2-\sqrt{3}}}{2-(2-\sqrt{3})}\)
\(=\frac{\sqrt{2}-\sqrt{2+\sqrt{3}}}{-\sqrt{3}}+\frac{\sqrt{2}+\sqrt{2-\sqrt{3}}}{-\sqrt{3}}\)
\(=\frac{2\sqrt{2}-\sqrt{2+\sqrt{3}}+\sqrt{2-\sqrt{3}}}{-\sqrt{3}}\)
Xét: \(A=\sqrt{2+\sqrt{3}}-\sqrt{2-\sqrt{3}}\)
\(A^2=4-2\sqrt{(2+\sqrt{3})(2-\sqrt{3})}=2\Rightarrow A=\sqrt{2}\)
\(P=\frac{2\sqrt{2}-A}{-\sqrt{3}}=\frac{2\sqrt{2}-\sqrt{2}}{-\sqrt{3}}=\frac{-\sqrt{2}}{\sqrt{3}}\)
