Cube Root Equations

Cubing undoes a cube root: ${\left(cbrt\left(u\right)\right)}^{3}$ is u. Take $cbrt\left(x+1\right) = -3$. Cube both sides: $x+1 = -27$, so x is -28. Check: $cbrt\left(-27\right)$ is -3, because ${\left(-3\right)}^{3} = -27$.

Notice the right side was negative, and that was fine. The cube root of a negative number is negative, so $cbrt\left(u\right) = -3$ is a sensible equation, unlike $\sqrt{u} = -3$.

Cubing never creates a false answer, because it keeps signs: 5 cubed is 125 and -5 cubed is -125. Two different real numbers never have the same cube, so if the cubes match, the numbers match. That is why no check is needed after cubing.