Definition:Cube Root

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Definition

Real Numbers

Let $x \in \R_{\ge 0}$ be a positive real number.

The cube roots of $x$ is the real number defined as:

$x^{\paren {1 / 3} } := \set {y \in \R: y^3 = x}$

where $x^{\paren {1 / 3} }$ is the $3$rd root of $x$.


The notation:

$y = \sqrt [3] x$

is usually encountered.


Complex Numbers

Definition:Cube Root/Complex Number

Examples

Cube Root of 2

The decimal expansion of the cube root of $2$ starts:

$\sqrt [3] 2 \approx 1 \cdotp 25992 \, 10498 \, 94873 \, 16476 \ldots$


Cube Root of 3

The decimal expansion of the cube root of $3$ starts:

$\sqrt [3] 3 \approx 1 \cdotp 44224 \, 95703 \, 07408 \, 3823 \ldots$


Cube Root of 5

The decimal expansion of the cube root of $5$ starts:

$\sqrt [3] 5 \approx 1 \cdotp 70997 \, 59466 \, 76696 \, 9893 \ldots$


Sources