The question is asking you to simplify the radical expression:

$\sqrt[3]{\frac{x^{12}}{y^9}}$

and write it in the form $A \cdot \sqrt[3]{B}$ or $A^3 \cdot \sqrt[3]{B}$.

Step-by-Step Solution:

1. Rewrite the expression: [ \sqrt[3]{\frac{x^{12}}{y^9}} = \frac{\sqrt[3]{x^{12}}}{\sqrt[3]{y^9}} ]

2. Simplify the cube roots:

   • For $\sqrt[3]{x^{12}}$: Recall the property $\sqrt[3]{a^n} = a^{n/3}$, so: $\sqrt[3]{x^{12}} = x^{12/3} = x^4$
   • For $\sqrt[3]{y^9}$: $\sqrt[3]{y^9} = y^{9/3} = y^3$

   So, the expression becomes: $\frac{x^4}{y^3}$

Final Answer:

The simplified expression is: $x^4 \cdot \frac{1}{y^3}$

Let me know if you have further questions or need additional explanation! Here's how to dive deeper:

5 Related Questions:

1. How do you simplify a fourth root or any higher roots of an expression with variables?
2. What is the general formula for simplifying radicals with exponents?
3. How would this problem change if it were a square root instead of a cube root?
4. Can we combine multiple radicals under the same root sign? If so, how?
5. What are some tips for rewriting expressions to match specific forms like $A \cdot \sqrt[3]{B}$?

Tip:

Always simplify powers within the radical first using fractional exponents—it often makes complex problems much easier!