The problem requires simplifying the radical expression:
$\sqrt[3]{8x^6}.$

The answer should be written in the form $A \sqrt[3]{B}$, $A^3 \sqrt[3]{B}$, or a similar form, where $A$ and $B$ are constants or expressions in $x$.

Step-by-Step Solution:

1. Break Down the Cube Root: [ \sqrt[3]{8x^6} = \sqrt[3]{8} \cdot \sqrt[3]{x^6}. ]

2. Simplify Each Part:

   • $\sqrt[3]{8} = 2$, because $2^3 = 8$.
   • $\sqrt[3]{x^6} = x^2$, because $x^2 \cdot x^2 \cdot x^2 = x^6$.

   So, the expression simplifies to: $2x^2.$

3. Final Answer: The simplified expression is: $2x^2.$

Let me know if you'd like further explanation or additional examples.

Relative Questions:

1. How do you simplify other cube roots involving both numbers and variables?
2. What happens if the exponent of $x$ is not divisible by 3?
3. How do square roots differ from cube roots in simplification?
4. Can cube roots ever result in absolute value terms?
5. How can we use properties of exponents to simplify radicals?

Tip: Always check if the exponents of variables are divisible by the radical’s index (e.g., 3 for cube roots) for easier simplifications.