Uniqueness of Positive Root of Positive Real Number

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Theorem

Positive Exponent

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 $y^n = x$.


Negative Exponent

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 $y^n = x$.