Existence and Uniqueness of Positive Root of Positive Real Number/Negative Exponent

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 $y^n = x$.