Evaluate:

$\sqrt[3]{-512}$

By prime factorisation:

$\sqrt[3]{512}=2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2$

$\quad=(2 \times 2 \times 2) \times(2 \times 2 \times 2) \times(2 \times 2 \times 2)$

$\sqrt[3]{-512}=-\sqrt[3]{(2 \times 2 \times 2)}=-8$

$\therefore \sqrt[3]{-512}=-(\sqrt[3]{512})=-8$