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Cube root

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Dec 15, 2021

In mathematics, a cube root of a number x is a number y such that y3 = x. All nonzero 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, denoted

83{displaystyle {sqrt[{3}]{8}}}

, is 2, because 23 = 8, while the other cube roots of 8 are

1+i3{displaystyle -1+i{sqrt {3}}}

and

1i3{displaystyle -1-i{sqrt {3}}}

. The three cube roots of −27i are

3i,33232i,and33232i.{displaystyle 3i,quad {frac {3{sqrt {3}}}{2}}-{frac {3}{2}}i,quad {text{and}}quad -{frac {3{sqrt {3}}}{2}}-{frac {3}{2}}i.}

Number whose cube is a given number

Plot of y = 3x. The plot is symmetric with respect to origin, as it is an odd function. At x = 0 this graph has a vertical tangent.
A unit cube (side = 1) and a cube with twice the volume (side = 32 = 1.2599… OEIS: A002580).

In some contexts, particularly when the number whose cube root is to be taken is a real number, one of the cube roots (in this particular case the real one) is referred to as the principal cube root, denoted with the radical sign

  3.{displaystyle {sqrt[{3}]{~^{~}}}.}

The cube root is the inverse function of the cube function if considering only real numbers, but not if considering also complex numbers: although one has always

(x3)3=x,{displaystyle left({sqrt[{3}]{x}}right)^{3}=x,}

the cube root of the cube of a number is not always this number. For example,

1+i3{displaystyle -1+i{sqrt {3}}}

is a cube root of 8, (that is,

(1+i3)3=8{displaystyle (-1+i{sqrt {3}})^{3}=8}

), but

1+i32=(1+i3)33.{displaystyle -1+i{sqrt {3}}neq 2={sqrt[{3}]{(-1+i{sqrt {3}})^{3}}}.}

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

y3=x. {displaystyle y^{3}=x. }

For any real number x, there is one real number y such that y3 = x. The cube function is increasing, so does not give the same result for two different inputs, and it covers all real numbers. In other words, it is a bijection, or one-to-one. Then we can define an inverse function that is also one-to-one. For real numbers, we can define a unique cube root of all real numbers. If this definition is used, the cube root of a negative number is a negative number.

The three cube roots of 1

If x and y are allowed to be complex, then 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. For instance, the cube roots of 1 are:

1,12+32i,1232i.{displaystyle 1,quad -{frac {1}{2}}+{frac {sqrt {3}}{2}}i,quad -{frac {1}{2}}-{frac {sqrt {3}}{2}}i.}

The last two of these roots lead to a relationship between all roots of any real or complex number. If a number is one cube root of a particular real or complex number, the other two cube roots can be found by multiplying that cube root by one or the other of the two complex cube roots of 1.

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