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$\sqrt{2m \sqrt[3]{\frac{1}{4m^2} \sqrt{\frac{n}{n}}} }: \sqrt[12]{nm} = \sqrt{2m\left (\frac{1}{4m^2}\cdot (\frac{n}{m})^{\frac{1}{2}}\right )^{\frac{1}{3}}} :(nm)^{\frac{1}{12}}=\\\\=\sqrt{\frac{2m\cdot n^{\frac{1}{6}}}{2^{\frac{2}{3}}\cdot m^\frac{2}{3}\cdot m^{\frac{1}{6}}}}}:(nm)^{\frac{1}{12}}=\sqrt{ \frac{2^{\frac{1}{3}}\cdot m^\frac{1}{3}n^{\frac{1}{6}}}{m^{\frac{1}{6}}}}:(nm)^{\frac{1}{12}}=$

$=\left(2^{\frac{1}{3}}\cdot m^{\frac{1}{6}}\cdot n^{\frac{1}{6}}\right )^{\frac{1}{2}}:(nm)^{\frac{1}{12}}=$ $\frac{2^{\frac{1}{6}}\cdot (nm)^{\frac{1}{12}}}{(nm)^{\frac{1}{12}}}=2^{\frac{1}{6}}=\sqrt[6]{2}$