Cube Root Calculator
Calculate cube roots with step-by-step solutions, perfect cube detection, complex roots visualization, and verification.
cbrt(x) = x^(1/3)Result
Cube Root
5
Enter a Number
Enter any number (including negatives)
Number of decimal places (2-15)
Display all 3 cube roots using De Moivre's theorem
3sqrt(125) = 5
Step-by-Step Solution
Given number: 125
Formula: cuberoot(x) = x^(1/3)
cuberoot(125) = 125^(1/3)
Result: 5
125 is a perfect cube!
5^3 = 125 = 125
Verification: (5)^3 = 125.0000000000 = 125
Result
Cube Root
5
?How Do You Calculate a Cube Root?
The cube root of x (written cbrt(x) or x^(1/3)) is the number that, when cubed, gives x. For example, cbrt(27) = 3 because 3 x 3 x 3 = 27. Unlike square roots, cube roots work with negative numbers: cbrt(-8) = -2. Every real number has exactly one real cube root, but three complex cube roots total (using De Moivre's theorem).
What is a Cube Root?
A cube root of a number x is a value y such that y^3 = x. Written as cbrt(x), the third root symbol, or x^(1/3). Unlike square roots, cube roots are defined for all real numbers, including negatives. Every number has exactly one real cube root and three cube roots total (including complex numbers), evenly distributed around the origin in the complex plane.
Key Facts About Cube Roots
- cbrt(x) is the number y where y^3 = x
- cbrt(27) = 3 because 3 x 3 x 3 = 27
- Cube roots of negative numbers are real: cbrt(-8) = -2
- Perfect cubes have integer cube roots: cbrt(125) = 5
- Every number has 3 cube roots (1 real, 2 complex)
- cbrt(0) = 0 and cbrt(1) = 1
- cbrt(a x b) = cbrt(a) x cbrt(b)
- The real cube root preserves sign: cbrt(-27) = -3
Frequently Asked Questions
A cube root of a number x is a value y such that y x y x y = x (y cubed equals x). For example, the cube root of 27 is 3 because 3 x 3 x 3 = 27. Unlike square roots, cube roots exist for all real numbers, including negatives.
Yes! Unlike square roots, cube roots of negative numbers are real. The cube root of -27 is -3, because (-3) x (-3) x (-3) = -27. This is because an odd number of negatives multiplied gives a negative result.
A perfect cube is a number whose cube root is an integer. Examples: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000. These are the cubes of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.
Every non-zero number has exactly three cube roots in the complex plane: one real root and two complex conjugate roots. The complex roots are found using De Moivre's theorem and are equally spaced 120 degrees apart on a circle.
De Moivre's theorem states that for a complex number in polar form (r, theta), its nth roots are found at angles (theta + 2*pi*k)/n for k = 0, 1, ..., n-1. This gives all n complex roots, equally distributed around a circle.
Last updated: 2025-01-15
Result
Cube Root
5