Odd Powers

To undo a power, take the matching root of both sides. For ${x}^{3} = -125$, take the cube root: x is $cbrt\left(-125\right)$, which is -5. Check: ${\left(-5\right)}^{3} = -125$.

An odd power keeps the sign of its base. A positive number cubed is positive and a negative one cubed is negative, so every real number is the cube of exactly one real number, and ${x}^{3} = -125$ has one real answer. The same holds for any odd power: ${x}^{5} = 32$ has only the real answer x = 2, since ${\left(-2\right)}^{5}$ is -32.

Taking an odd root never brings in a false answer, for the same reason cubing does not: an odd power keeps signs. So there is nothing to check.