\(=\left(3\sqrt{2}-2\sqrt{2}+\sqrt{14}\right).\sqrt{2}-\sqrt{7}\\ =\left(\sqrt{2}+\sqrt{14}\right).\sqrt{2}-\sqrt{7}\\ =2+2\sqrt{7}-\sqrt{7}\\ =2+\sqrt{7}\)

\(\frac{1}{3-\sqrt{7}}-\frac{1}{3+\sqrt{7}}=\frac{3+\sqrt{7}}{\left(3-\sqrt{7}\right)\left(3+\sqrt{7}\right)}-\frac{3-\sqrt{7}}{\left(3-\sqrt{7}\right)\left(3+\sqrt{7}\right)}\)

\(=\frac{3+\sqrt{7}-3+\sqrt{7}}{\left(3-\sqrt{7}\right)\left(3+\sqrt{7}\right)}=\frac{2\sqrt{7}}{9-7}=\sqrt{7}\)

a, \(\frac{1}{3-\sqrt{7}}-\frac{1}{3+\sqrt{7}}=\frac{3+\sqrt[]{7}-3+\sqrt{7}}{\left(3-\sqrt{7}\right)\left(3+\sqrt{7}\right)}\)

\(=\frac{2\sqrt{7}}{9-7}=\sqrt{7}\)

Tu bieu thuc \(\Leftrightarrow\frac{3.\left(\sqrt{7}+2\right)}{\left(\sqrt{7}-2\right)\left(\sqrt{7}+2\right)}-\frac{14\sqrt{7}}{7}+|\sqrt{7}-2|\)

                    \(\Leftrightarrow3\sqrt{7}+6-2\sqrt{7}+\sqrt{7}-2=2\sqrt{7}+4\)

Ta đặt: \(A=\sqrt{\sqrt{7}-\sqrt{3}}-\sqrt{\sqrt{7}+\sqrt{3}}\)

=> \(A^2=\left(\sqrt{\sqrt{7}-\sqrt{3}}-\sqrt{\sqrt{7}+\sqrt{3}}\right)^2\)

<=> \(A^2=\sqrt{7}-\sqrt{3}-2\sqrt{\left(\sqrt{7}-\sqrt{3}\right)\left(\sqrt{7}+\sqrt{3}\right)}+\sqrt{7}+\sqrt{3}\)

<=> \(A^2=2\sqrt{7}-2\sqrt{7-3}\)

<=> \(A^2=2\sqrt{7}-2\sqrt{4}=2\left(\sqrt{7}-2\right)\)

=> \(A=\sqrt{2\left(\sqrt{7}-2\right)}\)

Thay vào ta được:

\(\frac{\sqrt{2\left(\sqrt{7}-2\right)}}{\sqrt{\sqrt{7}-2}}=\sqrt{2}\)

\(\frac{4+\sqrt{2}-\sqrt{3}-\sqrt{6}+\sqrt{8}}{2+\sqrt{2}-\sqrt{3}}\)

\(=\frac{\left(\sqrt{4}+\sqrt{2}\right)-\left(\sqrt{3}+\sqrt{6}\right)+\left(\sqrt{4}+\sqrt{8}\right)}{2+\sqrt{2}-\sqrt{3}}\)  ( Tách 4 thành \(\sqrt{4}+\sqrt{4}\) )

\(=\frac{\sqrt{2}\left(\sqrt{2}+1\right)-\sqrt{3}\left(1+\sqrt{2}\right)+\sqrt{4}\left(1+\sqrt{2}\right)}{2+\sqrt{2}-\sqrt{3}}\)

\(=\frac{\left(\sqrt{2}-\sqrt{3}+2\right)\left(\sqrt{2}+1\right)}{2+\sqrt{2}-\sqrt{3}}\)

\(=\sqrt{2}+1\)

\(=\frac{2+\sqrt{2}-\sqrt{3}+2-\sqrt{6}+\sqrt{8}}{2+\sqrt{2}-\sqrt{3}}=1+\frac{\sqrt{2}\left(2+\sqrt{2}-\sqrt{3}\right)}{2+\sqrt{2}-\sqrt{3}}=1+\sqrt{2}\)

Ta có:

\(\frac{4+\sqrt{2}-\sqrt{3}-\sqrt{6}+\sqrt{8}}{2+\sqrt{2}-\sqrt{3}}\)

\(=\frac{2+\sqrt{2}-\sqrt{3}+2-\sqrt{6}+\sqrt{8}}{2+\sqrt{2}-\sqrt{3}}\)

\(=1+\frac{\sqrt{2}(2+\sqrt{2}-\sqrt{3})}{2+\sqrt{2}-\sqrt{3}}\)

\(=1+\sqrt{2}\)

Vậy \(\frac{4+\sqrt{2}-\sqrt{3}-\sqrt{6}+\sqrt{8}}{2+\sqrt{2}-\sqrt{3}}=1+\sqrt{2}\)

Ta có

\(\frac{4+\sqrt{2}-\sqrt{3}-\sqrt{6}+\sqrt{8}}{2+\sqrt{2}-\sqrt{3}}=\frac{2+\sqrt{2}-\sqrt{3}+2-\sqrt{6}+\sqrt{8}}{2+\sqrt{2}-\sqrt{3}}\)

\(=1+\frac{\sqrt{2}(2+\sqrt{2}-\sqrt{3})}{2+\sqrt{2}-\sqrt{3}}=1+\sqrt{2}\)

\(\text{Vậy }\frac{4+\sqrt{2}-\sqrt{3}-\sqrt{6}+\sqrt{8}}{2+\sqrt{2}-\sqrt{3}}=1+\sqrt{2}\)

\(\frac{\sqrt{3}+\sqrt{7}}{\sqrt{3}-\sqrt{7}}+\frac{\sqrt{3}-\sqrt{7}}{\sqrt{3}+\sqrt{7}}\)

\(=\frac{\left(\sqrt{3}+\sqrt{7}\right)\left(\sqrt{3}+\sqrt{7}\right)+\left(\sqrt{3}-\sqrt{7}\right)\left(\sqrt{3}-\sqrt{7}\right)}{\left(\sqrt{3}-\sqrt{7}\right)\left(\sqrt{3}+\sqrt{7}\right)}\)

\(=\frac{\left(\sqrt{3}+\sqrt{7}\right)^2+\left(\sqrt{3}-\sqrt{7}\right)^2}{3-7}\)

\(=\frac{3+2\sqrt{3}.\sqrt{7}+7+3-2\sqrt{3}.\sqrt{7}+7}{-4}\)

\(=\frac{3+7+3+7}{-4}\)

\(=\frac{20}{-4}=-5\)

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