Sign of Odd Power

Theorem

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

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

Then:

$x^n = 0 \iff x = 0$

$x^n > 0 \iff x > 0$

$x^n < 0 \iff x < 0$

That is, the sign of an odd power matches the number it is a power of.

Corollary

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

Then:

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

Proof

If $n$ is an odd integer, then $n = 2 k + 1$ for some $k \in \N$.

Thus $x^n = x \cdot x^{2 k}$.

But $x^{2 k} \ge 0$ from Even Power is Non-Negative.

The result follows.

$\blacksquare$

Sources

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