Even Power of Negative Real Number

Theorem

Let $x \in \R$ be a real number.

Let $n \in \Z$ be an even integer.

Then:

$\paren {-x}^n = x^n$

Proof

From Real Numbers form Ordered Field, $\R$ is a field.

By definition, $\R$ is therefore a ring.

The result follows from Power of Ring Negative.

$\blacksquare$