[tex]\bf \left( a^{-\frac{1}{4}}\cdot b^2 \right)^{\frac{5}{4}}\impliedby \textit{first off, distribute the exponent}
\\\\\\
a^{-\frac{1}{4}\cdot \frac{5}{4}}\cdot b^{2\cdot \frac{5}{4}}\implies a^{-\frac{5}{16}}\cdot b^{\frac{5}{2}}\implies a^{-\frac{5}{16}}\cdot b^{\frac{40}{16}}\implies \cfrac{b^{\frac{40}{16}}}{a^{\frac{5}{16}}}
\\\\\\
\cfrac{\sqrt[16]{b^{40}}}{\sqrt[16]{a^5}}\implies \sqrt[16]{\cfrac{b^{40}}{a^5}}[/tex]
now.. not sure if that's really simplified per se, I guess is a little.
you could also just take some "b" from the root and rationalize the denominator and end up with [tex]\bf \cfrac{b^2\sqrt[16]{a^{11}b^8}}{a}[/tex]
then again, not sure how simplified per se that one is either.
