If
\(f(x) = \begin{cases} -\sqrt[3]x & \text{if } x \geq 0,\\ x^2& \text{if } x <0. \end{cases} \)
f(f(f(f(512))))
\(\begin{align*} f(f(f(f(512)))) &=f(f(f(-8)))\\ &=f(f(64))\\ &=f(-4)\\ &=\boxed{16}. \end{align*}\)
