Let's simplify the given expression step by step:

$\frac{3y^{1/4}}{4x^{-2/3} \cdot y^{3/2} \cdot 3y^{1/2}}$

Step 1: Simplify the coefficients

The coefficient in the numerator is $3$, and in the denominator, we have $4 \times 3 = 12$. So, we can simplify the expression to:

$\frac{y^{1/4}}{12 \cdot x^{-2/3} \cdot y^{3/2} \cdot y^{1/2}}$

Step 2: Combine the powers of $y$

Now, let's deal with the powers of $y$. In the numerator, we have $y^{1/4}$, and in the denominator, we have $y^{3/2} \cdot y^{1/2}$.

Using the property of exponents $y^a \cdot y^b = y^{a + b}$, we combine the powers of $y$ in the denominator:

$y^{3/2} \cdot y^{1/2} = y^{(3/2 + 1/2)} = y^2$

So, the expression now becomes:

$\frac{y^{1/4}}{12 \cdot x^{-2/3} \cdot y^2}$

Step 3: Simplify the powers of $y$

Now, subtract the exponent of $y$ in the denominator from the exponent of $y$ in the numerator using the property $\frac{y^a}{y^b} = y^{a - b}$:

$y^{1/4 - 2} = y^{1/4 - 8/4} = y^{-7/4}$

So, the expression becomes:

$\frac{y^{-7/4}}{12 \cdot x^{-2/3}}$

Step 4: Simplify the powers of $x$

We can now deal with $x^{-2/3}$. Since it's in the denominator with a negative exponent, it moves to the numerator with a positive exponent:

$x^{2/3} \cdot y^{-7/4}$

Thus, the final simplified expression is:

$\frac{x^{2/3} y^{-7/4}}{12}$

This is the simplified form of the given expression.

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Would you like further details or explanations on any of the steps?

Related Questions:

1. How do you apply the properties of exponents to simplify expressions?
2. Can you explain how to deal with negative exponents in an expression?
3. What happens when you divide terms with the same base but different exponents?
4. How do you simplify expressions with multiple variables?
5. What are the steps to combine terms in a fraction involving both variables and exponents?

Tip:

Always keep track of the coefficients and exponents separately, simplifying each step methodically to avoid mistakes.