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Simplificar ala minina expresión y expresar sin exponentes negativos 

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

\[
\left( \frac{ay}{x} \right)^{\frac{1}{2}} \times \left( \frac{bx}{y} \right)^{\frac{1}{3}} \times \left( \frac{y^2}{a^2b^4} \right)^{\frac{1}{4}}
\]

### Step 1: Apply the exponents to each term inside the parentheses

1. For the first term:
   \[
   \left( \frac{ay}{x} \right)^{\frac{1}{2}} = \frac{(ay)^{\frac{1}{2}}}{x^{\frac{1}{2}}} = \frac{\sqrt{ay}}{\sqrt{x}}
   \]

2. For the second term:
   \[
   \left( \frac{bx}{y} \right)^{\frac{1}{3}} = \frac{(bx)^{\frac{1}{3}}}{y^{\frac{1}{3}}} = \frac{\sqrt[3]{bx}}{\sqrt[3]{y}}
   \]

3. For the third term:
   \[
   \left( \frac{y^2}{a^2b^4} \right)^{\frac{1}{4}} = \frac{(y^2)^{\frac{1}{4}}}{(a^2b^4)^{\frac{1}{4}}} = \frac{y^{\frac{1}{2}}}{a^{\frac{1}{2}}b}
   \]

### Step 2: Multiply the simplified expressions

Now we multiply these three simplified expressions together:

\[
\frac{\sqrt{ay}}{\sqrt{x}} \times \frac{\sqrt[3]{bx}}{\sqrt[3]{y}} \times \frac{y^{\frac{1}{2}}}{a^{\frac{1}{2}}b}
\]

This can be rewritten as:

\[
\frac{\sqrt{ay} \times \sqrt[3]{bx} \times y^{\frac{1}{2}}}{\sqrt{x} \times \sqrt[3]{y} \times a^{\frac{1}{2}}b}
\]

### Step 3: Combine the terms

Combine the terms in the numerator and the denominator:

**Numerator:**
- Combine the \(y\) terms: \(\sqrt{y} \times y^{\frac{1}{2}} = y^{\frac{1}{2} + \frac{1}{2}} = y\)
- Combine the \(a\) terms: \(\sqrt{a} = a^{\frac{1}{2}}\)
- Combine the \(b\) terms: \(\sqrt[3]{bx} \times b^{-1} = \frac{\sqrt[3]{bx}}{b}\)

So, the numerator simplifies to:
\[
a^{\frac{1}{2}} \times y \times \sqrt[3]{bx}
\]

**Denominator:**
- Combine the \(x\) terms: \(\sqrt{x} \times \sqrt[3]{x} = x^{\frac{1}{2} + \frac{1}{3}} = x^{\frac{5}{6}}\)
- Combine the \(y\) terms: \( \sqrt[3]{y} \)

So, the denominator simplifies to:
\[
x^{\frac{5}{6}} \times \sqrt[3]{y} \times a^{\frac{1}{2}}b
\]

### Step 4: Final Simplification

The final expression is:

\[
\frac{a^{\frac{1}{2}} \times y \times \sqrt[3]{bx}}{x^{\frac{5}{6}} \times \sqrt[3]{y} \times a^{\frac{1}{2}}b}
\]

Notice that \(a^{\frac{1}{2}}\) cancels out:

\[
\frac{y \times \sqrt[3]{bx}}{x^{\frac{5}{6}} \times \sqrt[3]{y} \times b}
\]

Now let's further simplify:

\[
\frac{y}{\sqrt[3]{y}} \times \frac{\sqrt[3]{bx}}{x^{\frac{5}{6}} \times b} = \frac{y^{\frac{3}{3} - \frac{1}{3}}}{x^{\frac{5}{6}} \times b} \times \sqrt[3]{bx}
\]

This gives us:

\[
\frac{y^{\frac{2}{3}} \times \sqrt[3]{bx}}{x^{\frac{5}{6}} \times b}
\]

Thus, the fully simplified expression is:

\[
\frac{y^{\frac{2}{3}} \times \sqrt[3]{bx}}{b \times x^{\frac{5}{6}}}
\]

### Final Expression:
\[
\boxed{\frac{y^{\frac{2}{3}} \cdot \sqrt[3]{bx}}{b \cdot x^{\frac{5}{6}}}}
\]

This is the simplified form without negative exponents.

Would you like more details on any part of the simplification, or do you have any questions?

### Related Questions:
1. How do you simplify expressions with fractional exponents?
2. What are the properties of exponents used in simplifying expressions?
3. Can you explain the steps in simplifying a radical expression?
4. How can negative exponents be converted to positive exponents?
5. What is the process for multiplying and dividing expressions with exponents?

### Tip:
When simplifying expressions with exponents, always apply exponent rules carefully, combining like terms wherever possible to simplify the expression step by step.

Learn how to simplify expressions with fractional exponents step-by-step. This detailed guide covers techniques for simplifying complex expressions involving fractional exponents and eliminating negative exponents. Suitable for high school students and anyone interested in algebraic simplification.

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