Let's simplify the given expression step by step. The expression is:

$\frac{\sqrt{(3^2 x^2 y)^3}}{\sqrt[3]{x^3}}$

Step 1: Simplify the numerator

The numerator is:

$\sqrt{(3^2 x^2 y)^3}$

Using the property of radicals, $\sqrt{a^b} = a^{b/2}$, we rewrite this as:

$(3^2 x^2 y)^3 = 3^{2 \cdot 3} x^{2 \cdot 3} y^3 = 3^6 x^6 y^3$

Now apply the square root:

$\sqrt{3^6 x^6 y^3} = 3^{6/2} x^{6/2} y^{3/2} = 3^3 x^3 y^{3/2}$

So the numerator simplifies to:

$27 x^3 y^{3/2}$

Step 2: Simplify the denominator

The denominator is:

$\sqrt[3]{x^3}$

Using the property of radicals, $\sqrt[n]{a^b} = a^{b/n}$, we rewrite this as:

$x^{3/3} = x^1 = x$

Step 3: Combine the simplified parts

Now we combine the simplified numerator and denominator:

$\frac{27 x^3 y^{3/2}}{x}$

Simplify the $x$-terms:

$27 x^{3-1} y^{3/2} = 27 x^2 y^{3/2}$

Final Answer:

The simplified expression is:

$27 x^2 y^{3/2}$

Do you have any questions or need further clarification?
Here are 5 related questions to expand your understanding:

1. How do properties of exponents apply when simplifying radical expressions?
2. What happens if the base in the numerator and denominator are different?
3. Can you rewrite $y^{3/2}$ as a radical expression?
4. How does the order of operations affect simplifying expressions with radicals?
5. What would the simplification be if the denominator was $\sqrt[4]{x^4}$ instead?

Tip: When working with radicals, always try converting them to fractional exponents for easier manipulation!