We are asked to simplify the expression:

$b^5 \sqrt{\frac{b^{10}}{4 b^4}}$

Step-by-Step Simplification:

1. Simplify the square root term: $\sqrt{\frac{b^{10}}{4 b^4}}$

   First, simplify the fraction inside the square root:

   $\frac{b^{10}}{4 b^4} = \frac{b^{10}}{b^4} \times \frac{1}{4} = b^{10 - 4} \times \frac{1}{4} = \frac{b^6}{4}$

   Now we have:

   $\sqrt{\frac{b^6}{4}} = \frac{\sqrt{b^6}}{\sqrt{4}} = \frac{b^3}{2}$

2. Substitute this back into the original expression:

   Now the expression becomes:

   $b^5 \times \frac{b^3}{2}$

3. Combine the powers of $b$: $b^5 \times b^3 = b^{5 + 3} = b^8$

4. Final simplified expression: $\frac{b^8}{2}$

Thus, the simplified expression is:

$\frac{b^8}{2}$

Final Answer:

$\boxed{\frac{b^8}{2}}$

Would you like any further explanation or clarification?

Here are some related questions:

1. How do you simplify square roots with variables in the numerator and denominator?
2. What are the rules for simplifying exponents when multiplying powers of the same base?
3. What would happen if $b$ was less than or equal to 0 in this expression?
4. How do you handle expressions involving fractional powers (e.g., square roots)?
5. Can you simplify expressions with multiple square roots or cube roots?

Tip: Always simplify the fraction inside a square root before extracting the root, as it can make the process easier.