Cube root

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In mathematics, the cube root ([\sqrt[3] ] ) of a number is the number which, when cubed (multiplied by itself and then multiplied by itself again), gives back the original number. For example, the cube root of 8 is 2, because 2 × 2 × 2 = 8, or:

[\sqrt[3] = 2.]

The cube root operation is associative with exponentiation and distributive with multiplication and division, but is not associative or distributive with addition or subtraction.

Contents

1 Formal definition
2 Infinitely nested cube roots
3 Cube root on standard calculator

3.1 Why this method works

4 See also
5 References

Formal definition

The cube roots of a number x are the numbers y which solve the equation

[y^3 = x\,]

If x and y are real, there is a unique solution and so the cube root of a real number is sometimes defined by this equation. If this definition is used, the cube root of a negative number is a negative number. The principle cube root of x is also represented by

[\sqrt[3] = x^]

If x and y are allowed to be complex, there are three solutions (if x is non-zero) and so x has three cube roots. A real number has one real cube root and two further cube roots, which form a complex conjugate pair. This can lead to some interesting results.

For instance, the cube roots of the number one are:

[1]
[-1 + \sqrti\over2]
[-1 - \sqrti\over2.]

These two roots lead to a relationship between all roots. If a number is one cube root of any real or complex number, the other two cube roots can be found by multiplying that number by the two complex cube roots of one.

For complex numbers the principle cube root is usually defined by

[x^ = \exp(\over3})]

where ln(x) is the principal branch of the natural logarithm. If we write x as

[x = r \exp(i \theta)]

where r is a non-negative real number and θ lies in the range

[-\pi < \theta \le \pi],

then the complex cube root is

[\sqrt[3] = \sqrt[3]\exp(i\theta/3)].

This means that in polar coordinates, we are taking the cube root of the radius and dividing the polar angle by three in order to define a cube root. With this definition, the cube root of a negative number is a complex number, and for instance [\sqrt[3]] will not be [-2], but rather [1 + i\sqrt].

In programs that are aware of the imaginary plane (such as Mathematica) are told to graph the cube root of x on a real number plane, they will not display any output for negative values of x. When a person wants negative values of the cube displayed, these programs must be explicitly told to only use real numbers. (In Mathematica, this can be achieved by executing the following line >>Miscellaneous`RealOnly`.)

Infinitely nested cube roots

Under certain conditions infinitely nested radicals such as

[ x = \sqrt[3]}}} ]

represent rational numbers. This rational number can be found by realizing that x also appears under the radical sign, which gives the equation

[ x = \sqrt[3]. ]

If we solve this equation, we find that x = 2. More generally, we find that

[ \sqrt[3]}}}] is the real root of the equation [ x^3-x-n=0 \,\!] for all n where n>0.

The same procedure also works to get

[ \sqrt[3]}}} ] is the real root of the equation [ x^3+x-n=0 \,\!] for all n where n>0.

Cube root on standard calculator

From the identity:

[\frac = \frac \left(1 + \frac\right) \left(1 + \frac\right) \left(1 + \frac\right) \left(1 + \frac}\right) \dots],

there is a simple method to compute cube roots using a non-scientific calculator, using only the multiplication and square root buttons, after the number is on the display. No memory is required.

Press the square root button once.
Press the multiplication button.
Press the square root button twice.
Press the multiplication button.
Press the square root button four times.
Press the multiplication button.
Press the square root button eight times.
Press the multiplication button...

One continues this process until the number does not change after pressing the multiplication button because the repeated square root gives 1 (this means that the solution has been figured to as many significant digits as the calculator can handle). Then, press the square root button one last time. At this point an approximation of the cube root of the original number will be shown in the display.

If the first multiplication is replaced by division, instead of the cube root, the fifth root will be shown on the display.

Why this method works

After raising x to the power in both sides of the above identity, one obtains:

[x^} = x^ \left(1 + \frac\right) \left(1 + \frac\right) \left(1 + \frac\right) \left(1 + \frac}\right) ...}] (*)

The left hand side is the cube root of x.

The steps shown in the method give:

After 2nd step:

[x^}]

After 4th step:

[x^ (1 + \frac)}]

After 6th step:

[x^ (1 + \frac) (1 + \frac)}]

After 8th step:

[x^ (1 + \frac) (1 + \frac) (1 + \frac)}]

etc.

After computing the necessary terms according to the calculator precision, the last square root finds the right hand of (*).

References