Existence and Uniqueness of Positive Root of Positive Real Number

Theorem

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

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

Then there always exists a unique $y \in \R: \paren {y \ge 0} \land \paren {y^n = x}$.

Hence the justification for the terminology the positive $n$th root of $x$ and the notation $x^{1/n}$.

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 $y \in \R: \paren {y \ge 0} \land \paren {y^n = x}$.

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 $y \in \R: \paren {y \ge 0} \land \paren {y^n = x}$.

$\blacksquare$