# Existence of Positive Root of Positive Real Number

## 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 $y^n = x$.