Cubic root

Calculator for computing a cubic root

Cubic root calculator

What is calculated?

This function returns the cubic root of the given argument (radicand). The argument may be a positive or negative real number.

Input values


Result
The result is shown with the selected number of decimal places

Cubic root info

Properties

Cubic root:

  • Inverse function of x³
  • Defined for all real numbers
  • Sign is preserved
  • Strictly increasing

Note: Unlike the square root, the radicand of a cubic root may be negative.

Examples
∛8 = 2
2³ = 8
∛27 = 3
3³ = 27
∛(-8) = -2
(-2)³ = -8
∛1 = 1
1³ = 1

Formula of the cubic root

General form
\[\sqrt[3]{x} = x^{1/3}\] Cubic root as a power
Definition
\[\sqrt[3]{x} = y \Leftrightarrow y^3 = x\] Inverse of the third power
Sign rule
\[\sqrt[3]{-x} = -\sqrt[3]{x}\] Sign is preserved
Arithmetic rules
\[\sqrt[3]{x \cdot y} = \sqrt[3]{x} \cdot \sqrt[3]{y}\] Product rule

Calculation example

Example: calculate ∛64

Given:

  • Radicand = 64
  • Sought: ∛64

Calculation:

\[\sqrt[3]{64} = 4\] \[\text{since } 4^3 = 4 \times 4 \times 4 = 64\]

Result: The cubic root of 64 is 4.

Example with negative radicand

Example: calculate ∛(-27)

Given:

  • Radicand = -27
  • Sought: ∛(-27)

Calculation:

\[\sqrt[3]{-27} = -3\] \[\text{since } (-3)^3 = (-3) \times (-3) \times (-3) = -27\]

Result: The cubic root of -27 is -3. The sign is preserved!

Definition of the cubic root

Difference to square root

Unlike the square root, the radicand may be a negative number, because (-3)^3 = (-3) · (-3) · (-3) = -27, while 3^3 = 3 · 3 · 3 = 27.

Sign rule

The sign of the root and the radicand are always identical. Negative numbers have negative cubic roots; positive numbers have positive cubic roots.

Mathematical properties
  • Domain: ℝ (all real numbers)
  • Range: ℝ (all real numbers)
  • Function: Strictly increasing
  • Symmetry: Odd function (point symmetric at origin)

Is this page helpful?            
Thank you for your feedback!

Sorry about that

How can we improve it?

Logarithmus und Exponenten

Log, Logarithmus zur Basis 10  •  Ln, Logarithmus zur Basis e  •  Logarithmus zur Basis a (Loga)  •  Exponential Funktion Basis 10x  •  Exponential Funktion Basis ex  •  Exponential Funktion Basis xy  •  Quadratwurzel  •  Kubikwurzel  •  N-te Wurzel  •  PowMod, Modular Exponentiation  •  Hypot  •