Existence of Positive Root of Positive Real Number
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Theorem
Let $x \in \R$ be a real number such that $x > 0$.
Let $n \in \Z$ be an integer such that $n \ne 0$.
Then there exists a $y \in \R: y \ge 0$ such that $y^n = x$.
Proof
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 exists a $y \in \R: y \ge 0$ 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 exists a $y \in \R: y \ge 0$ such that $y^n = x$.