[tex]a = \dfrac{}{}^{\sqrt 2 )}\dfrac{1}{(1+\sqrt 2-\sqrt 3)\sqrt 2};\quad b = \dfrac{1}{1+\sqrt 2+\sqrt 3}\\ \\ a = \dfrac{\sqrt 2}{2}\cdot \dfrac{1}{1+\sqrt 2-\sqrt 3};\quad b = \dfrac{1}{1+\sqrt 2+\sqrt 3}\\ \\ \\ a\cdot b = \dfrac{\sqrt 2}{2}\cdot \dfrac{1}{\big[(1+\sqrt 2) - \sqrt 3\big]\cdot \big[(1+\sqrt 2) +\sqrt 3\big]}\\ \\ a\cdot b = \dfrac{\sqrt 2}{2}\cdot \dfrac{1}{(1+\sqrt 2)^2 - \sqrt 3^2} \\ \\ a\cdot b =\dfrac{\sqrt 2}{2}\cdot \dfrac{1}{1+2\sqrt 2 + 2 - 3}[/tex]
[tex]a\cdot b = \dfrac{\sqrt 2}{2}\cdot \dfrac{1}{2\sqrt 2} = \dfrac{1}{4}\\ \\ \Rightarrow \mathrm{m.g.} =\sqrt{a\cdot b} = \sqrt{\dfrac{1}{4}} = \boxed{\dfrac{1}{2}}[/tex]
