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:

$\forall x \in \R: x^n = x^{2 k} = \paren {x^k}^2$

But from Square of Real Number is Non-Negative:

$\forall x \in \R: \paren {x^k}^2 \ge 0$

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

The result follows from Solution to Quadratic Equation.

$\blacksquare$

Sources

• 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 1$: Real Numbers: Exercise $\S 1.12 \ (1)$