Why do square roots come in pairs but cubed roots don't?

Answer 1

The square of a negative number is the same as the square of its positive counterpart, aka its additive inverse ( [-2]2 = 22 = 4), so every positive number has two square roots, a positive one and negative one (both 2 and -2 are square roots of 4). However, the cube of any number will always have the same sign as the original number (23 = 8, [-2]3 = -8). This all follows from simple arithmetic with signs. The product of any two negative numbers is positive, as is the product of any two positive numbers, while the product of a negative number and a positive number is negative. All squares, by definition, are the product of either two positive numbers or two negative numbers, and in either case, the product must be positive. But a cube is the product of a number and its square (x3 = x * x * x = x2 * x). But we already know that the square must be positive, whether original number is positive or negative. So the sign of the original number determines the sign of the cube (because a positive number times a positive number is positive and a positive number times a negative number is negative). If you apply that rule in reverse, then the sign of the cube root must be the same as the sign of the number you are taking the cube root of.

Think of it this way. If you are trying to calculate the square root of a number, y, you are looking for another number, x, for which it is true that x * x = y. For any positive number y, there are always two values of x that satisfy that equation, with one being positive and the other being negative, but both having the same absolute value. And therefore, every positive number has two square roots.

On the other hand, if you are trying to find the cube root of a number, y, you are looking for a number, z, for which it is true that z * z * z = y. For any number, y, either positive or negative, there will be only one value of z that satisfies that equation. Therefore, every number, positive or negative, has just one cube root.

Actually, technically, once you get into higher mathematics, what is really going on is that every number has 3 cube roots, but they all just happen to have the same value. In fact, for any "degree" of root (square root, cube root, 4th root, 5th root, ... 100th root, ...) the number of roots of a number is exactly equal to the degree of the root (a number will have 4 4th roots, 5 5th roots, 10 10th roots, 99 99th roots, etc.) But, if the degree of the root is odd, then all of the roots will have the same value, while if the degree is even, the roots will be evenly split between two values that are the additive inverses of each other. For example, the 5th roots of -243 are -3, -3, -3, -3, and -3, while the 6th roots of 64 are 2, 2, 2, -2, -2, and -2.

Note also that negative numbers cannot have any roots of any even degree (square roots, 4th roots, 6th roots, etc.) Actually, even that's not true when you get into really advanced math. Even negative numbers have even-degree roots, it's just that the roots are not real numbers. They are "imaginary" numbers. This is, I'm sure, way beyond your level of education in mathematics, and I'm not trying to confuse you. But if I hadn't included these last two paragraphs, some wise-guy mathematician would come along and "correct" me, and in the process probably confuse you even more. For your purposes, however, just ignore the last two paragraphs.