Let's simplify the given expression step by step.

The expression is:

[ \frac{\sqrt[3]{27x^6y^{12}}}{\sqrt[4]{x^8}} ]

Step 1: Simplify the numerator $\sqrt[3]{27x^6y^{12}}$

1. Simplify the constant:
   $27$ can be rewritten as $3^3$, so: [ \sqrt[3]{27} = \sqrt[3]{3^3} = 3 ]

2. Simplify the variable $x^6$:
   Using the property $\sqrt[n]{a^m} = a^{\frac{m}{n}}$, we get: $\sqrt[3]{x^6} = x^{\frac{6}{3}} = x^2$

3. Simplify the variable $y^{12}$:
   Similarly, for $y^{12}$: $\sqrt[3]{y^{12}} = y^{\frac{12}{3}} = y^4$

Thus, the numerator simplifies to: $3x^2y^4$

Step 2: Simplify the denominator $\sqrt[4]{x^8}$

For the denominator: $\sqrt[4]{x^8} = x^{\frac{8}{4}} = x^2$

Step 3: Simplify the entire expression

Now, the expression becomes: $\frac{3x^2y^4}{x^2}$

Here, $x^2$ in the numerator and denominator cancel each other out, leaving: $3y^4$

Final Answer

The simplified expression is:

$3y^4$

Would you like more details or have any questions? Here are five related questions you might find useful:

1. How do you simplify expressions involving different roots (e.g., cube root, fourth root)?
2. What is the general method to simplify radicals with variables?
3. How do you handle expressions where the roots involve fractions?
4. Can the process of canceling terms in a fraction ever lead to incorrect results?
5. What are the common mistakes to avoid when simplifying expressions with exponents?

Tip: Always look for common factors in the numerator and denominator to simplify expressions efficiently.