Cube Root Calculator

An online calculator to calculate the cube root of a real number. The definition and formulas for the cube roots of a number are given below.

Use of Cube Root Calculator

1 - Enter a real number x (-5.5 is the default value) in decimal form and the number of decimal places (positive integer) desired and press "Calculate Cube Root".

x =
Decimal Places =
Real Root =
Complex Root (1) = + i     
Complex Root (2) = + i     

Real Cube Root of a Real Number

A number y is the cube root of a number x if \( y^3 = x \).
Numerical examples
\( 1 \) is the cube root of 1 because \( 1^3 = 1 \)
\( 2\) is the cube root of 8 because \( 2^3 = 8 \)
\(- 10\) is the cube root of - 1000 because \((- 10)^3 = - 1000 \)
The cube root function notation is as follows

\( y = \sqrt[3]x \)

The graph of the cube root functions is shown below.

The graph of the cube root function

Complex Cube Root of a Real Number

Up to this point, the cube root is a real number. If we allow the cube root to be a complex number, then according to Demoivre's theorem there are three cube roots, one real and two complex conjugate, for any non zero real x and these roots are:
one real root given by
\( y_1 = \sqrt[3]x \)
two complex conjugate given by
\( y_2 = \sqrt[3]x ( - \dfrac{1}{2} + i \dfrac{\sqrt3}{2}) \)
\( y_3 = \sqrt[3]x ( - \dfrac{1}{2} - i \dfrac{\sqrt3}{2}) \)
where \(i\) is the imaginary unit such that \( i = \sqrt{-1} \).
Example
The three cube roots of - 27 are
\( y_1 = \sqrt[3]{-27} = - 3 \)
\( y_2 = \sqrt[3]{-27} ( - \dfrac{1}{2} + i \dfrac{\sqrt3}{2}) = \dfrac{3}{2} - \dfrac{3\sqrt3}{2}\)
\( y_3 = \sqrt[3]{-27} ( - \dfrac{1}{2} - i \dfrac{\sqrt3}{2}) = \dfrac{3}{2} + \dfrac{3\sqrt3}{2} \)

More References and Links

Cube Root Functions
De Moivre's Theorem Power and Root
Power Calculator .