Question

[Như]

Tính

\(\frac{1}{\sqrt{8}+\sqrt{7}}+\sqrt{175}-\frac{6\sqrt{2}-4}{3-\sqrt{2}}\)

\(\frac{\sqrt{6-\sqrt{11}}}{\sqrt{22}-\sqrt{2}}+\frac{6}{\sqrt{2}}-\frac{3}{\sqrt{2}+1}\)

Answer 1

[Trương Minh Trọng]

* \(\frac{1}{\sqrt{8}+\sqrt{7}}+\sqrt{175}-\frac{6\sqrt{2}-4}{3-\sqrt{2}}\)\(=\frac{\left(\sqrt{8}-\sqrt{7}\right)}{\left(\sqrt{8}+\sqrt{7}\right)\left(\sqrt{8}-\sqrt{7}\right)}+\sqrt{25.7}-\frac{2\sqrt{2}\left(3-\sqrt{2}\right)}{3-\sqrt{2}}\)

\(=\sqrt{8}-\sqrt{7}+5\sqrt{7}-2\sqrt{2}=2\sqrt{2}+4\sqrt{7}-2\sqrt{2}=4\sqrt{7}\)

** \(\frac{\sqrt{6-\sqrt{11}}}{\sqrt{22}-\sqrt{2}}+\frac{6}{\sqrt{2}}-\frac{3}{\sqrt{2}+1}\)\(=\frac{\sqrt{2}\sqrt{6-\sqrt{11}}}{\sqrt{2}\left(\sqrt{22}-\sqrt{2}\right)}+\frac{6\sqrt{2}}{2}-\frac{3\left(\sqrt{2}-1\right)}{\left(\sqrt{2}+1\right)\left(\sqrt{2}-1\right)}\)

\(=\frac{\sqrt{12-2\sqrt{11}}}{2\sqrt{11}-2}+3\sqrt{2}-\frac{3\sqrt{2}-3}{1}\)\(=\frac{\sqrt{\left(\sqrt{11}\right)^2-2.\sqrt{11}+1^2}}{2\left(\sqrt{11}-1\right)}+3\sqrt{2}-3\sqrt{2}+3\)

\(=\frac{\sqrt{11}-1}{2\left(\sqrt{11}-1\right)}+3=\frac{1}{2}+3=\frac{7}{2}\).

Answer 2

[Như]

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