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The given expression is:

\[
\frac{\frac{x^3 y^{1/2}}{5 \cdot 3}}{x^2 y^2 \cdot y^{-1/3}}
\]

### Step-by-step simplification:

1. **Simplify the numerator**:  
   The numerator is \(\frac{x^3 y^{1/2}}{15}\) (since \(5 \times 3 = 15\)).

2. **Simplify the denominator**:  
   In the denominator, we have:
   \[
   x^2 y^2 \cdot y^{-1/3} = x^2 y^{2 - 1/3} = x^2 y^{5/3}
   \]

3. **Rewrite the expression**:
   Now, the expression becomes:
   \[
   \frac{\frac{x^3 y^{1/2}}{15}}{x^2 y^{5/3}}
   \]

4. **Simplify the fraction**:
   \[
   = \frac{x^3 y^{1/2}}{15 \cdot x^2 y^{5/3}}
   \]

5. **Simplify the powers of \(x\) and \(y\)**:
   For \(x\), subtract the exponents:
   \[
   x^{3 - 2} = x^1
   \]
   For \(y\), subtract the exponents:
   \[
   y^{1/2 - 5/3} = y^{\frac{3}{6} - \frac{10}{6}} = y^{-\frac{7}{6}}
   \]

   So the expression becomes:
   \[
   \frac{x}{15 y^{7/6}}
   \]

### Final simplified expression:
\[
\frac{x}{15 y^{7/6}}
\]

Would you like further clarification or have additional questions? Here are some related questions:

1. How do you simplify expressions with fractional exponents?
2. What is the process for dividing variables with exponents?
3. How do you combine like terms in algebraic fractions?
4. Can you explain how negative exponents affect the simplification process?
5. What rules govern the simplification of rational expressions?

**Tip**: Always double-check your work by breaking down exponents step by step to avoid mistakes!

Simplify the expression: (x^3 * y^(1/2)) / (5 * 3 * x^2 * y^2 * y^(-1/3))

Learn how to simplify the algebraic expression (x^3 * y^(1/2)) / (5 * 3 * x^2 * y^2 * y^(-1/3)) using the rules of exponents. This step-by-step solution covers simplifying fractional exponents, combining like terms, and working with rational expressions. Suitable for high school students in Grades 9-10.

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