To simplify the given radical expression, we need to rewrite it in the required form. The expression is:

$\sqrt[3]{64x^{14}}$

Step 1: Break the expression inside the radical into factors

$64x^{14} = (64) \cdot (x^{14}).$

Step 2: Simplify the cube root of $64$

Since $64 = 4^3$, we have: $\sqrt[3]{64} = 4.$

Step 3: Simplify the cube root of $x^{14}$

Using the property of exponents for cube roots, $\sqrt[3]{x^{14}} = x^{14/3}$. Break this into: $x^{14/3} = x^4 \cdot x^{2/3},$ where $x^4$ is the integer part, and $x^{2/3}$ remains under the cube root.

Step 4: Combine terms

The simplified form of the expression is: $4x^4 \sqrt[3]{x^2}.$

Step 5: Apply absolute value if needed

When dealing with even powers and cube roots, we don't typically need the absolute value for the root part. Hence: [ A = 4|x^4|, \quad B = x^2, \quad \text{and } A\sqrt[3]{B} = 4x^4\sqrt[3]{x^2}. ]

Final Answer:

$\boxed{4x^4 \sqrt[3]{x^2}}$

Would you like a breakdown of cube root properties, or need help with other similar problems? Here are five related questions to practice:

1. Simplify $\sqrt[3]{27x^9}$ into its simplest form.
2. How would you simplify $\sqrt[3]{8x^{10}}$?
3. Can you rewrite $\sqrt[3]{125x^{12}}$ in the form $A \sqrt[3]{B}$?
4. What are the general rules for breaking down cube roots of powers?
5. How does the presence of fractional exponents affect cube roots?

Tip: For radicals with even powers, remember to check if the absolute value symbol is necessary based on the context!