Question

Every number has ______ cube root. It can be __________ or ___________.

A) One cube root, which is a real number.
B) One cube root, which can be a real number or a complex number.
C) Two cube roots, one of which is a real number and the other is a complex number.
D) Three cube roots, two of which are complex numbers.

Answer 1

Every number has exactly one real cube root, which is a real number. For any real number x, there is a unique real number y such that y^3 = x. Cube roots have a single solution in the real numbers, unlike square roots which may have two.

Step-by-step explanation:

The correct answer is A) One cube root, which is a real number. Every real number has exactly one real cube root. This means that for any real number x, there is one and only one real number y such that y3 = x. For example, the cube root of 27 is 3 because 33 = 27.

Unlike square roots, where there are typically two solutions (a positive and a negative root), cube roots have a single solution in the real number system, and it can be positive, negative, or zero. For instance, the cube root of -8 is -2 since (-2)3 = -8.

When working with cube roots in the context of equilibrium problems or analyzing cubic equations, it's crucial to recognize that cube roots provide real solutions.

This stands in contrast to square roots, which may sometimes yield complex solutions when dealing with negative radicands. Understanding how to find cube roots is essential, as is knowing how to perform these operations on a calculator if necessary.