Cube root
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Image:Cube root.png In mathematics, the cube root (<math>\sqrt[3]{ } </math> ) of a number is a number which, when cubed (multiplied by itself and then multiplied by itself again), gives back the original number. For example, a cube root of 8 is 2, because 2 × 2 × 2 = 8, or:
- <math>\sqrt[3]{8} = 2.</math>
The cube root operation is associative with exponentiation and distributive with multiplication and division, but is not associative or distributive with addition or subtraction.
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Formal definition
Formally, the cube root of a real or complex number x is a real or complex solution y to the equation:
- <math>y^3 = x\,</math>
which leads to the equivalent notation for the cube root as:
- <math>y = x^{1\over3}.</math>
A non-zero complex number has three cube roots. A real number has a unique real cube root, but when treated as a complex number it has two further cube roots, which are complex conjugates of each other. This can lead to some interesting results.
For instance, the cube roots of the number one (often referred to as unity in this context) are:
- <math>1</math>
- <math>-1 + \sqrt{3}i\over2</math>
- <math>-1 - \sqrt{3}i\over2.</math>
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.
When treated purely as a real function of a real variable, we may define a real cube root for all real numbers by setting:
- <math>(-x)^{1\over3} = -x^{1\over3}.</math>
This is correct only for the domain of real numbers: for complex numbers we define instead the cube root to be:
- <math>x^{1\over3} = \exp({\ln{x}\over3})</math>
where ln(x) is the principal branch of the natural logarithm. If we write x as :
- <math>x = r \exp(i \theta)</math>
where r is a non-negative real number and θ lies in the range
- <math>-\pi < \theta \le \pi</math>,
then the complex cube root is
- <math>\sqrt[3]{x} = \sqrt[3]{r}\exp(i\theta/3)</math>.
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. Hence, for instance, <math>\sqrt[3]{-8}</math> will not be <math>-2</math>, but rather <math>1 + i\sqrt{3}</math>, a somewhat counterintuitive result.
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
- <math> x = \sqrt[3]{6+\sqrt[3]{6+\sqrt[3]{6+\sqrt[3]{6+\cdots}}}} </math>
represent rational numbers. This rational number can be found by realizing that x also appears under the radical sign, which gives the equation
- <math> x = \sqrt[3]{6+x}. </math>
If we solve this equation, we find that x = 2. More generally, we find that
- <math> \sqrt[3]{n+\sqrt[3]{n+\sqrt[3]{n+\sqrt[3]{n+\cdots}}}}</math> is the real root of the equation <math> x^3-x-n=0 \,\!</math> for all n where n>0.
The same procedure also works to get
- <math> \sqrt[3]{n-\sqrt[3]{n-\sqrt[3]{n-\sqrt[3]{n-\cdots}}}} </math> is the real root of the equation <math> x^3+x-n=0 \,\!</math> for all n where n>0.
Cube root on standard calculator
From the identity:
- <math>\frac{1}{3} = \frac{1}{2^2} \left(1 + \frac{1}{2^2}\right) \left(1 + \frac{1}{2^4}\right) \left(1 + \frac{1}{2^8}\right) \left(1 + \frac{1}{2^{16}}\right) \dots</math>,
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:
- <math>x^{\frac{1}{3}} = x^{\frac{1}{2^2} \left(1 + \frac{1}{2^2}\right) \left(1 + \frac{1}{2^4}\right) \left(1 + \frac{1}{2^8}\right) \left(1 + \frac{1}{2^{16}}\right) ...}</math> (*)
The left hand side is the cube root of x.
The steps shown in the method give:
After 2nd step:
- <math>x^{\frac{1}{2}}</math>
After 4th step:
- <math>x^{\frac{1}{2} (1 + \frac{1}{2^2})}</math>
After 6th step:
- <math>x^{\frac{1}{2} (1 + \frac{1}{2^2}) (1 + \frac{1}{2^4})}</math>
After 8th step:
- <math>x^{\frac{1}{2} (1 + \frac{1}{2^2}) (1 + \frac{1}{2^4}) (1 + \frac{1}{2^8})}</math>
etc.
After computing the necessary terms according to the calculator precision, the last square root finds the right hand of (*).
See also
External links
de:Kubikwurzel fr:Racine cubique lt:Kubinė šaknis nl:Derdemachtswortel pl:Pierwiastek sześcienny fi:Kuutiojuuri