We aim to simplify the radical expression $\sqrt[3]{64x^{14}}$ and write it in the form $A \cdot \sqrt[3]{B}$, where $A$ and $B$ are constants or expressions in $x$.

Step-by-Step Solution

1. Break down the terms inside the radical:

   • $64x^{14}$ can be factored as: $64x^{14} = 64 \cdot x^{12} \cdot x^2$ Here, $x^{12}$ is a perfect cube, and $64 = 4^3$ is also a perfect cube.

2. Separate perfect cubes from the rest:

   • Rewrite the expression using these factorizations: [ \sqrt[3]{64x^{14}} = \sqrt[3]{64} \cdot \sqrt[3]{x^{12}} \cdot \sqrt[3]{x^2} ]
   • Simplify the perfect cube terms: [ \sqrt[3]{64} = 4, \quad \sqrt[3]{x^{12}} = x^4 ]

3. Combine the simplified terms:

   • We now have: [ \sqrt[3]{64x^{14}} = 4x^4 \cdot \sqrt[3]{x^2