\frac{8^{\frac{1}{2}}*\sqrt[3]{4*}\sqrt[3]{2}}{4*\sqrt[3]{16*}\sqrt[4]{2}}=
=\frac{(2^3)^{\frac{1}{2}}*(2^2)^{\frac{1}{3}}*2^{\frac{1}{3}}}{2^2*(2^4)^{\frac{1}{3}}*2^{\frac{1}{4}}}=
=\frac{2^{\frac{3}{2}}*2^{\frac{2}{3}}*2^{\frac{1}{3}}}{2^2*2^{\frac{4}{3}}*2^{\frac{1}{4}}}=
=\frac{2^{\frac{9+4+2}{6}}}{2^{\frac{24+16+3}{12}}}=\frac{2^{\frac{15}{6}}}{2^{\frac{43}{12}}}=
=\frac{2^{\frac{30}{12}}}{2^{\frac{43}{12}}}=2^{\frac{30-43}{12}}=2^{-\frac{13}{12}}