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Simplificar ala minina expresión y expresar sin exponentes negativos 

To simplify the given expression and express it without negative exponents, we'll go through the following steps.

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

### Step 1: Simplify each factor

#### First term:
\[
\left(\frac{ay}{x}\right)^{\frac{1}{2}} = \frac{(ay)^{\frac{1}{2}}}{x^{\frac{1}{2}}} = \frac{\sqrt{ay}}{\sqrt{x}} = \frac{\sqrt{a}\sqrt{y}}{\sqrt{x}}
\]

#### Second term:
\[
\left(\frac{bx}{y}\right)^{\frac{1}{3}} = \frac{(bx)^{\frac{1}{3}}}{y^{\frac{1}{3}}} = \frac{\sqrt[3]{bx}}{\sqrt[3]{y}}
\]

#### Third term:
\[
\left(\frac{y^2}{a^2b^4}\right)^{\frac{1}{4}} = \frac{\left(y^2\right)^{\frac{1}{4}}}{\left(a^2b^4\right)^{\frac{1}{4}}} = \frac{y^{\frac{2}{4}}}{a^{\frac{2}{4}}b^{\frac{4}{4}}} = \frac{\sqrt{y}}{\sqrt{a}b}
\]

### Step 2: Combine the simplified terms
Now, we multiply the three simplified expressions together:

\[
\frac{\sqrt{a}\sqrt{y}}{\sqrt{x}} \times \frac{\sqrt[3]{bx}}{\sqrt[3]{y}} \times \frac{\sqrt{y}}{\sqrt{a}b}
\]

### Step 3: Combine and simplify
First, cancel out and simplify where possible:
- \(\sqrt{a}\) and \(\sqrt{a}\) cancel out.
- \(b\) from the third term cancels with the \(b\) in the second term.

So we get:
\[
\frac{\sqrt{y}}{\sqrt{x}} \times \frac{\sqrt[3]{x}}{\sqrt[3]{y}} \times \frac{\sqrt{y}}{b}
\]

Further simplification gives:
\[
\frac{y^{\frac{1}{2}}y^{\frac{1}{2}}}{x^{\frac{1}{2}}b} = \frac{y}{x^{\frac{1}{2}}b}
\]

Now multiply the remaining factors:
- Combine exponents for \(y\) and \(x\).

Finally, the simplified expression is:
\[
\frac{y}{b\sqrt{x}}
\]

### Final Simplified Expression:
\[
\boxed{\frac{y}{b\sqrt{x}}}
\]

This is the expression simplified to the minimum without any negative exponents.

Would you like further details or have any questions? Here are some related questions to consider:

1. How would you handle the expression if there were additional variables in the numerator or denominator?
2. What techniques can be used to simplify expressions involving higher roots?
3. How can you convert expressions involving roots back into exponent form?
4. What happens if the variables \(a\), \(b\), or \(x\) are negative? 
5. How would the expression change if we had a higher degree root, like \(\sqrt[5]{x}\)?

**Tip:** When simplifying expressions with exponents, always check if terms can be canceled out or combined before performing the operations to make the process easier.

Learn how to simplify an expression and express it without negative exponents with a step-by-step solution. This problem covers exponents, roots, and simplification techniques, suitable for high school students.

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