Existence and Uniqueness of Positive Root of Positive Real Number/Negative Exponent
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Theorem
Let $x \in \R$ be a real number such that $x \ge 0$.
Let $n \in \Z$ be an integer such that $n < 0$.
Then there always exists a unique $y \in \R: \paren {y \ge 0} \land \paren {y^n = x}$.
Proof
Proof of Existence
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 exists a $y \in \R: y \ge 0$ such that $y^n = x$.
Proof of Uniqueness
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 $y^n = x$.