Even Power is Non-Negative

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

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

Then $x^n \ge 0$.

That is, all even powers are positive.

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

Then $n = 2 k$ for some $k \in \Z$.

Thus:
 * $\forall x \in \R: x^n = x^{2k} = \left({x^k}\right)^2$

But from Square of Real Number is Non-Negative:
 * $\forall x \in \R: \left({x^k}\right)^2 \ge 0$

and so there is no real number whose square is negative.

The result follows from Solution to Quadratic Equation.