1. The problem asks us to find the cube root of 512, which means finding a number $x$ such that $x^3 = 512$. 2. We start by finding prime factors of 512. Since $512 = 2^9$, we have that $512 = 2^9$. 3. The cube root of $512$ is the same as the cube root of $2^9$, which can be written as $\sqrt[3]{2^9}$. 4. Using the property of roots and exponents, $\sqrt[3]{2^9} = 2^{9/3} = 2^3$. 5. Calculate $2^3 = 8$. Therefore, $\boxed{8}$ is the cube root of 512.