Simplify the Nested Radical: ∛(√(64x¹²))

To solve the problem $\sqrt[3]{\sqrt{64 \cdot x^{12}}}$, follow these detailed steps:

• Step 1: Simplify the inner expression.
  The expression inside the radical is $64 \cdot x^{12}$.
• Step 2: Simplify the inner square root.
  

  First, we need to find $\sqrt{64 \cdot x^{12}}$.

  The square root of a product can be expressed as the product of the square roots: $\sqrt{64} \cdot \sqrt{x^{12}}$.

  Simplifying further, we find:

  • $\sqrt{64} = 8$, since $8^2 = 64$.
  • $\sqrt{x^{12}} = x^{6}$, because $(x^{6})^2 = x^{12}$.

  Thus, the inner square root becomes $8x^6$.

• Step 3: Simplify using the cube root.
  

  Next, apply the cube root to the result of the inner square root: $\sqrt[3]{8x^6}$.

  The cube root of a product can also be expressed as the product of the cube roots:

  • $\sqrt[3]{8} = 2$, since $2^3 = 8$.
  • $\sqrt[3]{x^6} = x^{6/3} = x^{2}$, because $(x^2)^3 = x^6$.

  Thus, the expression simplifies to $2x^2$.

Therefore, the solution to this problem is $2x^2$, which corresponds to choice 2 in the provided options.