Cube root
Definition
A cube root of a number x is a number y such that y3 = x. In other words, it is a value that, when cubed (multiplied by itself twice), gives the original number. All real numbers have exactly one real cube root and a pair of complex conjugate cube roots, and all nonzero complex numbers have three distinct complex cube roots. For example, the real cube root of 8 is 2, because 23 = 8. The term is derived from the mathematical operation of finding this root, which is akin to the square root operation but with a cubed (rather than squared) result.
Mathematical Properties
The cube root operation is an example of a broader class of operations known as radical or root operations. Like all root operations, it is the inverse of an exponentiation operation – in this case, the operation of raising to the power of three.
The cube root of a product or quotient is equal to the product or quotient of their cube roots. This property is a consequence of the laws of exponents.
The cube root function is an odd function, meaning it is symmetric with respect to the origin. This property is a consequence of the fact that the cube of any negative number is negative.
The cube root function is monotonically increasing, meaning it never decreases as its input increases. This property is a consequence of the fact that the cube of any number is greater than the cube of any smaller number.
Calculation Methods
There are several methods for calculating cube roots, some of which are suitable for manual computation and others of which are used in computer algebra systems.
The simplest method for finding the cube root of a number is to use the guess-and-check method. This method involves making an initial guess for the cube root, cubing it to see how close the result is to the original number, and then adjusting the guess based on the result. This process is repeated until the cube of the guess is sufficiently close to the original number.
Another method for finding cube roots is the method of successive approximations, also known as the Newton-Raphson method. This method involves making an initial guess for the cube root, calculating the error in this guess, and then adjusting the guess based on the error. This process is repeated until the error is sufficiently small.
A more advanced method for finding cube roots is the method of continued fractions. This method involves representing the cube root as a continued fraction and then calculating successive approximations to this fraction.
Applications
Cube roots have applications in various fields of science and engineering. In physics, they appear in formulas for the volume of a sphere and the moment of inertia of a solid body. In chemistry, they appear in the ideal gas law and the van der Waals equation. In computer science, they are used in algorithms for sorting and searching data.