a. \(\sqrt{\dfrac{5+2\sqrt{6}}{5-2\sqrt{6}}}+\sqrt{\dfrac{5-2\sqrt{6}}{5+2\sqrt{6}}}\)
\(=\sqrt{\dfrac{\left(5+2\sqrt{6}\right)^2}{\left(5-2\sqrt{6}\right)\left(5+2\sqrt{6}\right)}}+\sqrt{\dfrac{\left(5-2\sqrt{6}\right)^2}{\left(5+2\sqrt{6}\right)\left(5-2\sqrt{6}\right)}}\)
\(=\sqrt{\dfrac{\left(5+2\sqrt{6}\right)^2}{5^2-24}}+\sqrt{\dfrac{\left(5-2\sqrt{6}\right)^2}{5^2-24}}\)
\(=\sqrt{\left(5+2\sqrt{6}\right)^2}+\sqrt{\left(5-2\sqrt{6}\right)^2}\)
\(=\left|5+2\sqrt{6}\right|+\left|5-2\sqrt{6}\right|\)
\(=5+2\sqrt{6}+5-2\sqrt{6}=10\)
b. Giải tương tự
+ \(\sqrt{\dfrac{5+2\sqrt{6}}{5-2\sqrt{6}}}+\sqrt{\dfrac{5-2\sqrt{6}}{5+2\sqrt{6}}}\)
= \(\sqrt{\left(5+2\sqrt{6}\right)\times\left(5+2\sqrt{6}\right)}+\sqrt{\left(5-2\sqrt{6}\right)\times\left(5-2\sqrt{6}\right)}\)
= \(\sqrt{\left(5+2\sqrt{6^2}\right)}+\sqrt{\left(5-2\sqrt{6^2}\right)}\)
\(5+2\sqrt{6}+5-2\sqrt{6}\)
\(=5+5\)
\(=10\)
+ \(\sqrt{\dfrac{3+\sqrt{5}}{3-\sqrt{5}}}+\sqrt{\dfrac{3-\sqrt{5}}{3+\sqrt{5}}}\)
\(=\sqrt{\dfrac{\left(3+\sqrt{5}\right)\times\left(3+\sqrt{5}\right)}{4}}+\sqrt{\dfrac{\left(3-\sqrt{5}\right)\times\left(3-\sqrt{5}\right)}{4}}\)
\(=\sqrt{\dfrac{(3+\sqrt{5)}^2}{4}}+\sqrt{\dfrac{\left(3-\sqrt{5}\right)^2}{4}}\)
\(=\dfrac{3+\sqrt{5}}{2}+\dfrac{3-\sqrt{5}}{2}\)
\(=\dfrac{3+\sqrt{5}+3-\sqrt{5}}{2}\)
\(=\dfrac{3+3}{2}\)
\(=\dfrac{6}{2}=3\)
