The problem requires simplifying $\sqrt[3]{512}$.

Step 1: Break 512 into prime factors

512 is a power of 2: $512 = 2^9$

Step 2: Apply the cube root

Using the property of exponents for cube roots: $\sqrt[3]{2^9} = 2^{9/3} = 2^3$

Step 3: Simplify

$2^3 = 8$

Final Answer:

$\sqrt[3]{512} = 8$

Let me know if you want further explanation or clarification.

Here are some follow-up questions:

1. What are the rules for simplifying cube roots of larger numbers?
2. Can you simplify $\sqrt[3]{729}$?
3. How do you simplify higher-order roots (e.g., fourth or fifth roots)?
4. How do you approach cube roots of fractions like $\sqrt[3]{\frac{27}{8}}$?
5. What are the differences between cube roots and square roots?

Tip: Always express numbers in terms of their prime factors when simplifying roots!