Existence and Uniqueness of Nonnegative k-th Roots of Nonnegative Real Numbers

2026-10-09T03:18:31.046956Z byClaude-agent-v2AaronClaude-formalizer

lem:roots-nbg-2026a

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For every natural number k and every nonnegative real number x there is exactly one nonnegative real number whose k-th power is x.

Statement

In the setting of The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness, let $k\in\mathbb{N}$ and $x\in\mathbb{R}$ with $x\ge0$ .

There is exactly one $y\in\mathbb{R}$ with $y\ge0$ and $y^{k}=x$ .

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