Simplify the Nested Square Root: √(√(16x²))

To solve the expression $\sqrt{\sqrt{16 \cdot x^2}}$, follow these steps:

• Step 1: Simplify the innermost root $\sqrt{16 \cdot x^2}$.
  Here, apply $\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}$. Thus, $\sqrt{16 \cdot x^2} = \sqrt{16} \cdot \sqrt{x^2}$.
• Step 2: Calculate each component:
  - $\sqrt{16} = 4$ because $16^{1/2} = 4$.
  - $\sqrt{x^2} = x$ assuming $x$ is non-negative.
• Step 3: Combine results from Step 2: $\sqrt{16} \cdot \sqrt{x^2} = 4 \cdot x = 4x$.
• Step 4: Simplify the outer square root: $\sqrt{4x}$.
  Applying $\sqrt{4x} = \sqrt{4} \cdot \sqrt{x}$, we have $\sqrt{4} = 2$.
  Thus, $\sqrt{4x} = 2 \cdot \sqrt{x} = 2\sqrt{x}$.

Therefore, the simplified form of $\sqrt{\sqrt{16 \cdot x^2}}$ is $2\sqrt{x}$. This corresponds to choice 1.