\(\sqrt{\dfrac{5+2\sqrt{6}}{5-2\sqrt{6}}}-\sqrt{\dfrac{5-2\sqrt{6}}{5+2\sqrt{6}}}+\sqrt{15-6\sqrt{6}}\)
\(=\sqrt{\dfrac{\left(\sqrt{3}\right)^2+2\sqrt{3}.\sqrt{2}+\left(\sqrt{2}\right)^2}{\left(\sqrt{3}\right)^2-2\sqrt{3}.\sqrt{2}+\left(\sqrt{2}\right)^2}}-\sqrt{\dfrac{\left(\sqrt{3}\right)^2-2\sqrt{3}.\sqrt{2}+\left(\sqrt{2}\right)^2}{\left(\sqrt{3}\right)^2+2\sqrt{3}.\sqrt{2}+\left(\sqrt{2}\right)^2}}+\\ \sqrt{\left(\sqrt{6}\right)^2-2.\sqrt{6}.3+3^2}\)
\(=\sqrt{\dfrac{\left(\sqrt{3}+\sqrt{2}\right)^2}{\left(\sqrt{3}-\sqrt{2}\right)^2}}-\sqrt{\dfrac{\left(\sqrt{3}-\sqrt{2}\right)^2}{\left(\sqrt{3}+\sqrt{2}\right)^2}}+\sqrt{\left(\sqrt{6}-3\right)^2}\)
\(=\dfrac{\left|\sqrt{3}+\sqrt{2}\right|}{\left|\sqrt{3}-\sqrt{2}\right|}-\dfrac{\left|\sqrt{3}-\sqrt{2}\right|}{\left|\sqrt{3}+\sqrt{2}\right|}+\left|\sqrt{6}-3\right|\)
\(=\dfrac{\sqrt{3}+\sqrt{2}}{\sqrt{3}-\sqrt{2}}-\dfrac{\sqrt{3}-\sqrt{2}}{\sqrt{3}+\sqrt{2}}-\left(\sqrt{6}-3\right)\) (vì \(\left\{{}\begin{matrix}\sqrt{3}+\sqrt{2}>0\\\sqrt{3}-\sqrt{2}>0\\\sqrt{6}-3< 0\end{matrix}\right.\))
\(=\dfrac{\left(\sqrt{3}+\sqrt{2}\right)^2-\left(\sqrt{3}-\sqrt{2}\right)^2}{\left(\sqrt{3}-\sqrt{2}\right)\left(\sqrt{3}+\sqrt{2}\right)}-\left(\sqrt{6}-3\right)\)
\(=\dfrac{\left(\sqrt{3}+\sqrt{2}+\sqrt{3}-\sqrt{2}\right)\left(\sqrt{3}+\sqrt{2}-\sqrt{3}+\sqrt{2}\right)}{3-2}-\left(\sqrt{6}-3\right)\)
\(=2\sqrt{3}.2\sqrt{2}-\sqrt{6}+3\)
\(=4\sqrt{6}-\sqrt{6}+3=3\sqrt{6}+3\)
