Find all real roots: Cube roots of 512

The cube root of 512 is 8 because \(8^3 = 512\). However, since we're looking for all real roots, we can also consider complex roots. The cube roots of a number can be expressed in polar form. For 512, which can be written as \(512 = 512(\cos 0 + i\sin 0)\), the cube roots can be found using De Moivre's theorem. Thus, the real root is 8, and the two other cube roots are \(8 \left( \cos \frac{2\pi}{3} + i \sin \frac{2\pi}{3} \right)\) and \(8 \left( \cos \frac{4\pi}{3} + i \sin \frac{4\pi}{3} \right)\), which are -4 + 4i√3 and -4 - 4i√3. However, if we consider only real roots, the answer is simply 8. In summary, the only real cube root of 512 is 8, and it's quite handy to know that this means any real number raised to the third power has a single real solution. So, whenever you're faced with cube roots, remember—there's only one real root waiting for you, easily calculable!