\(\dfrac{12\sqrt{6}}{\sqrt{7+2\sqrt{6}}-\sqrt{7-2\sqrt{6}}}=\dfrac{12\sqrt{6}}{\sqrt{\left(\sqrt{6}+1\right)^2}-\sqrt{\left(\sqrt{6}-1\right)^2}}=\dfrac{12\sqrt{6}}{\sqrt{6}+1-\sqrt{6}+1}=\dfrac{12\sqrt{6}}{2}=6\sqrt{6}\)
\(=\dfrac{12\sqrt{6}}{\sqrt{6+2.\sqrt{6}.1+1}-\sqrt{6-2.\sqrt{6}.1+1}}\)\(=\dfrac{12\sqrt{6}}{\sqrt{6}+1-\sqrt{6}+1}=\dfrac{12\sqrt{6}}{2}=6\sqrt{6}\)
\(\dfrac{12\sqrt{6}}{\sqrt{7+2\sqrt{6}}-\sqrt{7-2\sqrt{6}}}\)
\(=\dfrac{12\sqrt{6}}{\sqrt{1^2+2\sqrt{6}+\left(\sqrt{6}\right)^2}-\sqrt{1^2-2\sqrt{6}+\left(\sqrt{6}\right)^2}}\)
\(=\dfrac{12\sqrt{6}}{\sqrt{\left(1+\sqrt{6}\right)^2}-\sqrt{\left(1-\sqrt{6}\right)^2}}\)
\(=\dfrac{12\sqrt{6}}{\left|1+\sqrt{6}\right|-\left|1-\sqrt{6}\right|}\)
\(=\dfrac{12\sqrt{6}}{1+\sqrt{6}-\left(1-\sqrt{6}\right)}\)
\(=\dfrac{12\sqrt{6}}{1+\sqrt{6}-1+\sqrt{6}}\)
\(=\dfrac{12\sqrt{6}}{2\sqrt{6}}=6\)
