Definition:Fifth Root
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Definition
Real Numbers
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.
Complex Numbers
Definition:Fifth Root/Complex Number
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:
- $\sqrt [5] 3 \approx 1 \cdotp 24573 \, 09396 \, 15517 \, 32596 \, 668 \ldots$