Find the cube-roots of -0.512

$\sqrt[3]{-0.512}\\[1em] = \sqrt[3]{-\dfrac{512}{1000}}\\[1em] = -\dfrac{\sqrt[3]{512}}{\sqrt[3]{1000}}$

Prime factors of 512:

$\therefore -\dfrac{\sqrt[3]{512}}{\sqrt[3]{1000}} = -\dfrac{\sqrt[3]{(2 \times 2\times 2)\times(2\times 2\times 2)\times (2\times 2\times 2)}}{\sqrt[3]{10 \times 10\times 10}}\\[1em] = -\dfrac{(2 \times 2 \times 2)}{10}\\[1em] = -\dfrac{8}{10}\\[1em] = - 0.8$

Hence, $\sqrt[3]{-0.512} = -0.8$