Simplify the Nested Cube Root: ∛(∛(512x^27))

To solve the given problem, we'll follow these steps:

• Step 1: Simplify the innermost cube root $\sqrt[3]{512x^{27}}$.
• Step 2: Simplify the next cube root $\sqrt[3]{(\cdot)}$ from the result of step 1.

Let's go through each step:

Step 1: Consider the expression $\sqrt[3]{512x^{27}}$.
First, evaluate $\sqrt[3]{512}$. Since $512 = 8^3$, we have $\sqrt[3]{512} = 8$.
For $\sqrt[3]{x^{27}}$, use the property $\sqrt[n]{a^m} = a^{m/n}$:
$\sqrt[3]{x^{27}} = x^{27/3} = x^9$.
Thus, $\sqrt[3]{512x^{27}} = 8x^9$.

Step 2: Now, evaluate the outer cube root $\sqrt[3]{8x^9}$.
$\sqrt[3]{8} = 2$ since $8 = 2^3$.
For $\sqrt[3]{x^9}$, again use the rule $\sqrt[n]{a^m} = a^{m/n}$:
$\sqrt[3]{x^9} = x^{9/3} = x^3$.
Therefore, $\sqrt[3]{8x^9} = 2x^3$.

In conclusion, the simplified expression is $2x^3$.

Thus, the solution to the problem is $2x^3$, which corresponds to choice 3.