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.

Discover how to simplify the expression 16a√(4a/√a) using algebraic methods. This solution covers key concepts in radicals and exponentiation, breaking down the process step by step. Ideal for students in Grades 10-12.

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