Cube Root

The cube root of a number $ n $ is any number $ x $ solution of the equation: $ x^3 = n $.

The cube root of $ n $ is denoted $ \sqrt[3]{n} $ or $ n^{1/3} $.

The cube root is the inverse operation of cubing.

Calculating a cubic root is not easy to do by hand except for usual values such as: $ \sqrt[3]{1} = 1 $, $ \sqrt[3]{8} = 2 $, $ \sqrt[3]{27} = 3 $, $ \sqrt[3]{64} = 4 $, $ \sqrt[3]{125} = 5 $, $ \sqrt[3]{1000} = 10 $

To calculate the complex roots of $ n $, solve the equation $ x^3 = n $ in $ \mathbb{C} $ which is equivalent to finding the complex roots of the degree 3 polynomial: $ x^3 - n = 0 $

On a calculator, use the exponent button and the formula: $ \sqrt[3]{x} = x^{1/3} $

Some calculators have a cube root key ∛ directly.

On a spreadsheet like Microsoft Excel, use the same formula as for a calculator, for a value in A1 write A1^(1/3) or POWER(A1;1/3)

The root simplifier will attempt to factor the expression under the root with a perfect cube.

Simplification can be done manually in steps:

— Decompose the number into prime factors.

— Identify the perfect cubes among these factors.

— Remove these perfect cubes from the radical by extracting their cube root.

A cubic number is the cube of an integer (cubed value).

If the cube root of a number $ x $ is an integer (relative, without decimal part), then $ x $ is a cubic number.

The first perfect cubes are:

Cube root of 1 is 1 because $ \sqrt[3]1 = 1^{\frac{1}{3}} = 1 $

However, in complex numbers, there are three cube roots of unity: $ 1 $, $ e^{2\pi i /3} = -\frac{1}{2} + i\frac{\sqrt{3}}{2} $ and $ e^{-2\pi i /3} = -\frac{1}{2} - i\frac{\sqrt{3}}{2} $

In some software, cbrt stands for cube root abbreviation cb of cube and rt for root, similar to sqrt for square root.

The Unicode standard proposes the symbol U+221B ∛

In LaTeX language, write \sqrt[3]{x}

In programming languages, write cbrt(x) or x**(1/3)