When tasked with finding the cube root of a negative number, you’re essentially seeking a number that, when cubed, yields the original negative figure. As an illustration, the cube root of $-8$ is $-2$, because $(-2) \times (-2) \times (-2) = -8$.

This operation primarily involves two steps. Initially, you disregard the negative sign and determine the cube root of the number as though it were positive. This is referred to as computing the cube root of the absolute value. For instance, to find the cube root of $-27$, you first compute the cube root of $27$. This is $3$, because $3 \times 3 \times 3 = 27$.

Subsequently, you reapply the negative sign to your result. Thus, the cube root of $-27$ is $-3$, which holds because $(-3) \times (-3) \times (-3) = -27$.

It’s crucial to remember that, unlike square roots, cube roots of negative numbers are real numbers. This is due to the fact that the product of three negative numbers is a negative number, thereby making it feasible to have a real cube root for negative values. This property is beneficial in various mathematical contexts and real-world applications, including solving equations and modeling physical phenomena.