Solve Nested Cube Roots: Simplifying ∛(∛512)

Solve the following exercise:

$\sqrt[3]{\sqrt[3]{512}}=$

To solve this problem, we'll proceed with the following steps:

• Step 1: Compute $\sqrt[3]{512}$.
  Given $512$, recognize that $512 = 2^9$ since $2^9 = 512$. Thus, we have:
• $\sqrt[3]{512} = \sqrt[3]{2^9} = 2^{9/3} = 2^3 = 8$.
• Step 2: Compute $\sqrt[3]{8}$.
  From Step 1, we found $\sqrt[3]{512} = 8$. Now find $\sqrt[3]{8}$:
• $\sqrt[3]{8} = \sqrt[3]{2^3} = 2^{3/3} = 2$.

Therefore, the solution to the expression $\sqrt[3]{\sqrt[3]{512}}$ is $2$.