The expression provided is:

$\sqrt{\frac{16x^7 y^8}{x^2 y}}$

We need to simplify this expression given that $x$ and $y$ are positive real numbers.

Step 1: Simplify the expression inside the square root

Start by simplifying the fraction under the square root:

$\frac{16x^7 y^8}{x^2 y} = 16 \cdot \frac{x^7}{x^2} \cdot \frac{y^8}{y}$

Simplify each part separately:

$\frac{x^7}{x^2} = x^{7-2} = x^5$

$\frac{y^8}{y} = y^{8-1} = y^7$

So, the expression under the square root becomes:

$16x^5y^7$

Step 2: Take the square root of the simplified expression

Now, take the square root of each part:

$\sqrt{16x^5y^7} = \sqrt{16} \cdot \sqrt{x^5} \cdot \sqrt{y^7}$

$\sqrt{16} = 4$

The square root of $x^5$ can be split into:

$\sqrt{x^5} = \sqrt{x^4 \cdot x} = \sqrt{x^4} \cdot \sqrt{x} = x^2 \cdot \sqrt{x}$

Similarly, the square root of $y^7$ can be split into:

$\sqrt{y^7} = \sqrt{y^6 \cdot y} = \sqrt{y^6} \cdot \sqrt{y} = y^3 \cdot \sqrt{y}$

Final Simplified Expression

Combining everything:

$4x^2 y^3 \sqrt{xy}$

So, the simplified expression is:

$\boxed{4x^2 y^3 \sqrt{xy}}$

Would you like any more details or have any questions?

Related Questions:

1. How do you simplify square roots involving variables with exponents?
2. What is the process of simplifying fractions with exponents in the numerator and denominator?
3. How do you handle square roots of products involving multiple variables?
4. Why is it important that $x$ and $y$ are positive real numbers in this context?
5. How do you deal with higher powers when simplifying square roots?

Tip:

When dealing with square roots of variables raised to powers, remember to break down the power into an even part (which you can fully take the square root of) and an odd part (which remains under the square root).