Uniqueness of Positive Root of Positive Real Number/Negative Exponent

Theorem

Let $x \in \R$ be a real number such that $x > 0$.

Let $n \in \Z$ be an integer such that $n < 0$.

Then there is at most one $y \in \R: y \ge 0$ such that $y^n = x$.

Proof

Let $m = -n$.

Let $g$ be the real function defined on $\hointr 0 \to$ defined by:

$\map g y = y^m$

From the definition of power:

$\map g y = \dfrac 1 {\map f y}$

Hence $\map g y$ is strictly decreasing.

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From Strictly Monotone Mapping with Totally Ordered Domain is Injective:

there is at most one $y \in \R: y \ge 0$ such that $y^n = x$.

$\blacksquare$