Existence of Positive Root of Positive Real Number/Negative Exponent

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Theorem

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

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

Let $m = -n$.

Then $m > 0$.

Let $g$ be the real function defined on $\left[{0 \,.\,.\, \to}\right)$ defined by:

$g \paren y = y^m$

Since $x \ge 0$:

$\dfrac 1 x \ge 0$

By Existence of Positive Root of Positive Real Number: Positive Exponent there is a $y > 0$ such that:

$g \paren y = \dfrac 1 x$

It follows from the definition of power that:

\(\displaystyle f \paren y\) \(=\) \(\displaystyle \dfrac 1 {g \paren y}\)
\(\displaystyle \) \(=\) \(\displaystyle \dfrac 1 {1 / x}\)
\(\displaystyle \) \(=\) \(\displaystyle x\)

$\blacksquare$