The given problem is to simplify the radical expression:

$\sqrt[3]{\frac{x^3}{y^6}}$

Step-by-step solution:

1. Break the expression inside the cube root into parts: [ \sqrt[3]{\frac{x^3}{y^6}} = \frac{\sqrt[3]{x^3}}{\sqrt[3]{y^6}}. ]

2. Simplify each term inside the cube root:

   • For $\sqrt[3]{x^3}$: The cube root of $x^3$ is $x$.
   • For $\sqrt[3]{y^6}$: Use the rule $\sqrt[3]{y^a} = y^{a/3}$. So: $\sqrt[3]{y^6} = y^{6/3} = y^2.$

3. Combine the results: [ \frac{\sqrt[3]{x^3}}{\sqrt[3]{y^6}} = \frac{x}{y^2}. ]

Final Answer:

The simplified expression is: $\boxed{\frac{x}{y^2}}$

Let me know if you'd like a deeper explanation! 😊

Related Questions:

1. How do cube roots differ from square roots in simplifying expressions?
2. What are the general properties of radicals involving fractional exponents?
3. Can you explain how to simplify $\sqrt[3]{x^5}$?
4. How do you rationalize denominators involving cube roots?
5. What happens if the numerator or denominator contains a negative exponent?

Tip:

When simplifying radical expressions, always look for ways to rewrite powers using fractional exponents—they can make the process more straightforward!