The given expression simplifies to:

$\left(\frac{b^2 \sqrt{b^5}}{a^{5/2}}\right)^{0.4}$

Now, we can simplify further by rewriting all terms with positive exponents:

$= \frac{b^{0.8}}{a^{2}}$

This is the fully simplified form of the expression with only positive exponents.

Would you like more details on how this simplification was done, or do you have any further questions?

Here are 5 related questions to deepen your understanding:

1. How do you simplify fractional exponents involving square roots?
2. What are the rules for handling exponents when multiplying and dividing terms?
3. Can fractional exponents always be rewritten as radicals? If so, how?
4. How do you apply exponentiation to an entire fraction?
5. What techniques are used to ensure all exponents in an expression are positive?

Tip: Always apply the rules of exponents carefully when handling expressions with multiple variables to avoid mistakes during simplification!