a)
(a^{\frac{5}{6}}*a^{\frac{1}{2}}):a^{\frac{1}{3}}=(a^{\frac{5}{6}+\frac{3}{6}}):a^{\frac{1}{3}}=(a^{\frac{8}{6}}):{a^{\frac{1}{3}}}=a^{\frac{4}{3}-\frac{1}{3}}=a^1=a
b)
(a^{-\frac{1}{2}}:a^{-\frac{3}{2}})^{0,4}=(a^{-\frac{1}{2}+\frac{3}{2}})^{0,4}=(a^1)^{0,4}=a^{0,4}
c)
{a^{-\frac{2}{3}}*\sqrt[3]{a^2}*\frac{1}{\sqrt[4]{a^3}}=a^{-\frac{2}{3}}*a^{\frac{2}{3}}*(a^{\frac{3}{4}})^{-1}=a^{-\frac{2}{3}+\frac{2}{3}}*a^{-\frac{3}{4}}=a^0*a^{-\frac{3}{4}}=1*a^{-\frac{3}{4}}=a^{-\frac{3}{4}}}
d)
\sqrt[8]{a^3*\sqrt[4]{a^3*\sqrt{a}}}=a^{\frac{3}{8}}*(a^{\frac{3}{4}})^{\frac{1}{8}}*((a^{\frac{1}{2}})^{\frac{1}{4}})^{\frac{1}{8}}=a^{\frac{3}{8}}*a^{\frac{3}{32}}*a^{\frac{1}{64}}=
=a^{\frac{24}{64}}*a^{\frac{6}{64}}*a^{\frac{1}{64}}=a^{\frac{31}{64}}