https://r2.erweima.ai/mathbot/image/17259276940656tpxrp3m.jpg

The given expression is:

\[
\frac{\sqrt[4]{x^3 y^2}}{\sqrt[4]{3xy}}
\]

We need to simplify this.

### Step-by-step simplification:

1. **Apply the property of division of radicals:**
   \[
   \frac{\sqrt[4]{x^3 y^2}}{\sqrt[4]{3xy}} = \sqrt[4]{\frac{x^3 y^2}{3xy}}
   \]
   
2. **Simplify the expression inside the fourth root:**
   \[
   \frac{x^3 y^2}{3xy} = \frac{x^3}{x} \cdot \frac{y^2}{y} = x^2 y
   \]
   So, we have:
   \[
   \sqrt[4]{\frac{x^3 y^2}{3xy}} = \sqrt[4]{\frac{x^2 y}{3}}
   \]

3. **Final expression:**
   \[
   \sqrt[4]{\frac{x^2 y}{3}}
   \]

This corresponds to option **a)** in the given choices.

### Answer:
The simplified expression is:

\[
\boxed{\text{a) } \frac{\sqrt[4]{x^2 y}}{3}}
\]

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

1. How do we handle more complex fractional exponents?
2. What is the difference between square roots and higher-order roots?
3. How can you simplify expressions involving radicals in the denominator?
4. What other methods can we use to simplify radical expressions?
5. How does rationalizing the denominator work for expressions like this?

**Tip:** When dealing with radical expressions, always look for common factors you can cancel or simplify inside the root.

La expresión \(\frac{\sqrt[4]{x^3 y^2}}{\sqrt[4]{3xy}}\) se reduce a:

Aprende a simplificar la expresión \(\frac{\sqrt[4]{x^3 y^2}}{\sqrt[4]{3xy}}\) paso a paso. Este problema cubre conceptos clave de la simplificación de radicales y las propiedades de los radicales, adecuado para estudiantes de grados 10-12.

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