Definition:Fifth Root/Real

Definition

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

The fifth root of $x$ is the real number defined as:

$x^{\paren {1 / 5} } := \set {y \in \R: y^5 = x}$

where $x^{\paren {1 / 5} }$ is the $5$th root of $x$.

The notation:

$y = \sqrt [5] x$

is usually encountered.

Examples

Fifth Root of $2$

The decimal expansion of the $5$th root of $2$ starts:

$\sqrt [5] 2 \approx 1 \cdotp 14869 \, 83554 \, 99703 \, 50067 \, 986 \ldots$

Fifth Root of $3$

The decimal expansion of the $5$th root of $3$ starts:

$\sqrt [5] 3 \approx 1 \cdotp 24573 \, 09396 \, 15517 \, 32596 \, 668 \ldots$