(3(y)^1/4) / [(4(x)^-2/3) x ((y)^3/2) x (3(y)^1/2)]

We are given the expression:

\[
\frac{3y^{\frac{1}{4}}}{4x^{-\frac{2}{3}} \cdot y^{\frac{3}{2}} \cdot 3y^{\frac{1}{2}}}
\]

### Step-by-step Simplification:

1. **Simplify the numerator and denominator**:
   - The numerator is \(3y^{\frac{1}{4}}\).
   - The denominator is \(4x^{-\frac{2}{3}} \cdot y^{\frac{3}{2}} \cdot 3y^{\frac{1}{2}}\).

2. **Combine like terms in the denominator**:
   - The constants \(4\) and \(3\) multiply to give \(12\).
   - The powers of \(y\) can be combined by adding the exponents:
     - \(y^{\frac{3}{2}} \cdot y^{\frac{1}{2}} = y^{\frac{3}{2} + \frac{1}{2}} = y^2\).
   - Thus, the denominator becomes \(12x^{-\frac{2}{3}} y^2\).

3. **Rewrite the expression**:
   \[
   \frac{3y^{\frac{1}{4}}}{12x^{-\frac{2}{3}} y^2}
   \]

4. **Simplify the constants**:
   \[
   \frac{3}{12} = \frac{1}{4}
   \]
   So, the expression is now:
   \[
   \frac{y^{\frac{1}{4}}}{4x^{-\frac{2}{3}} y^2}
   \]

5. **Simplify the powers of \(y\)**:
   Use the rule \(y^{a} / y^{b} = y^{a-b}\):
   \[
   y^{\frac{1}{4}} / y^2 = y^{\frac{1}{4} - 2} = y^{\frac{1}{4} - \frac{8}{4}} = y^{-\frac{7}{4}}
   \]
   Now the expression is:
   \[
   \frac{y^{-\frac{7}{4}}}{4x^{-\frac{2}{3}}}
   \]

6. **Simplify the powers of \(x\)**:
   Since \(x^{-\frac{2}{3}}\) is in the denominator, it becomes \(x^{\frac{2}{3}}\) in the numerator:
   \[
   \frac{x^{\frac{2}{3}} y^{-\frac{7}{4}}}{4}
   \]

### Final Answer:

\[
\frac{x^{\frac{2}{3}} y^{-\frac{7}{4}}}{4}
\]

This is the simplified form of the given expression.

Would you like more details or have any questions?

### 5 Related Questions:
1. How do you simplify expressions with negative exponents?
2. What are the rules for multiplying exponents?
3. How can you simplify expressions with multiple variables?
4. What is the process for combining terms with the same base in algebra?
5. How do fractional exponents relate to roots?

### Tip:
When simplifying algebraic expressions with exponents, always remember to combine powers of the same base by adding or subtracting exponents, and apply the rules of negative exponents to move terms between the numerator and denominator.

This page provides a detailed step-by-step solution for simplifying the expression (3(y)^1/4) / [(4(x)^-2/3) x ((y)^3/2) x (3(y)^1/2)], covering key algebraic concepts like exponent rules, product and quotient of powers, and simplifying negative exponents. Suitable for Grades 9-12 students.

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