The expression you've provided is $16a\sqrt{\frac{4a}{\sqrt{a}}}$.

Let's simplify this step by step:

Step 1: Simplify the inside of the square root

You have $\frac{4a}{\sqrt{a}}$.

Since $\sqrt{a}$ is the same as $a^{\frac{1}{2}}$, you can rewrite the expression as: $\frac{4a}{a^{\frac{1}{2}}}$

Step 2: Simplify the fraction

Subtract the exponents of $a$ in the numerator and the denominator: $\frac{4a^{1}}{a^{\frac{1}{2}}} = 4a^{1 - \frac{1}{2}} = 4a^{\frac{1}{2}} = 4\sqrt{a}$

Step 3: Substitute back into the original expression

Now, substitute $4\sqrt{a}$ back into the square root: $16a\sqrt{4\sqrt{a}}$

Step 4: Simplify the square root

We know that $\sqrt{4\sqrt{a}} = \sqrt{4} \cdot \sqrt{\sqrt{a}}$. Since $\sqrt{4} = 2$ and $\sqrt{\sqrt{a}} = a^{\frac{1}{4}}$, we get: $\sqrt{4\sqrt{a}} = 2a^{\frac{1}{4}}$

Step 5: Multiply the expression

Finally, multiply the entire expression: $16a \cdot 2a^{\frac{1}{4}} = 32a^{1 + \frac{1}{4}} = 32a^{\frac{5}{4}}$

So, the simplified expression is: $\boxed{32a^{\frac{5}{4}}}$

Would you like more details or have any questions?

Here are some related questions:

1. How would you simplify $\sqrt{\frac{a^2}{b}}$?
2. What is the result of multiplying $\sqrt{x}$ by $\sqrt{y}$?
3. How do you simplify $\sqrt[3]{8a^3}$?
4. Can you factor the expression $16a^{\frac{5}{4}}$?
5. What is the difference between rational and irrational exponents?

Tip: When simplifying expressions with square roots, always try to express all terms with the same base to make calculations easier.