Question

Tính GTBT:

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

Answer 1

\(B=\frac{\sqrt{2}\left(4+\sqrt{7}\right)}{6+\sqrt{8+2\sqrt{7}}}+\frac{\sqrt{2}\left(4-\sqrt{7}\right)}{6-\sqrt{8-2\sqrt{7}}}=\frac{\sqrt{2}\left(4+\sqrt{7}\right)}{6+\sqrt{\left(\sqrt{7}+1\right)^2}}+\frac{\sqrt{2}\left(4-\sqrt{7}\right)}{6-\sqrt{\left(\sqrt{7}-1\right)^2}}\)

\(=\frac{\sqrt{2}\left(4+\sqrt{7}\right)}{7+\sqrt{7}}+\frac{\sqrt{2}\left(4-\sqrt{7}\right)}{7-\sqrt{7}}=\frac{\sqrt{2}\left(4+\sqrt{7}\right)\left(7-\sqrt{7}\right)}{35}+\frac{\sqrt{2}\left(4-\sqrt{7}\right)\left(7+\sqrt{7}\right)}{35}\)

\(=\frac{21\sqrt{2}+3\sqrt{14}}{35}+\frac{21\sqrt{2}-3\sqrt{14}}{35}=\frac{42\sqrt{2}}{35}=\frac{6\sqrt{2}}{5}\)