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The given expression is:

\[
f(x, y) = \frac{9 \left( y \sqrt{x^3} \right)^3}{x \left( 3 \sqrt[3]{y} \right)^2}
\]

Let's simplify it step by step.

### Step 1: Simplify the numerator
In the numerator, we have:

\[
9 \left( y \sqrt{x^3} \right)^3
\]

First, recognize that \(\sqrt{x^3} = x^{3/2}\). So, the term becomes:

\[
y \sqrt{x^3} = y x^{3/2}
\]

Now, cube this expression:

\[
\left( y x^{3/2} \right)^3 = y^3 x^{9/2}
\]

Thus, the numerator simplifies to:

\[
9 y^3 x^{9/2}
\]

### Step 2: Simplify the denominator
In the denominator, we have:

\[
x \left( 3 \sqrt[3]{y} \right)^2
\]

Recognize that \(\sqrt[3]{y} = y^{1/3}\). So, this expression becomes:

\[
3 \sqrt[3]{y} = 3 y^{1/3}
\]

Now, square this expression:

\[
\left( 3 y^{1/3} \right)^2 = 9 y^{2/3}
\]

Thus, the denominator simplifies to:

\[
x \cdot 9 y^{2/3} = 9x y^{2/3}
\]

### Step 3: Combine the simplified terms
Now, the entire expression becomes:

\[
\frac{9 y^3 x^{9/2}}{9 x y^{2/3}}
\]

We can cancel out the common factor of 9:

\[
\frac{y^3 x^{9/2}}{x y^{2/3}}
\]

### Step 4: Simplify further
Now simplify the powers of \(x\) and \(y\):

For \(x\), we have \(x^{9/2}\) divided by \(x^1\), which simplifies to:

\[
x^{9/2 - 1} = x^{7/2}
\]

For \(y\), we have \(y^3\) divided by \(y^{2/3}\), which simplifies to:

\[
y^{3 - 2/3} = y^{9/3 - 2/3} = y^{7/3}
\]

### Final Simplified Expression:
\[
f(x, y) = x^{7/2} y^{7/3}
\]

Would you like further clarification or a detailed explanation on any part of this? 

Here are five related questions:
1. How do you simplify fractional exponents in algebraic expressions?
2. Can you explain how to simplify terms involving cube roots and square roots together?
3. How does the process of canceling terms work in expressions with exponents?
4. What are the rules for exponentiation when multiplying and dividing powers of the same base?
5. How do you handle exponents in the numerator and denominator of a fraction?

**Tip:** When simplifying expressions involving roots and exponents, always convert roots into fractional exponents to make the simplification process easier.

Simplify the expression 9(y√x³)³ / x(3³√y)²

Learn how to simplify the expression 9(y√x³)³ / x(3³√y)² step by step, involving the use of fractional exponents and exponentiation rules. This problem covers key concepts like algebraic simplifications and exponent rules, ideal for high school students in Grades 10-12.

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