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

If $n \in \Z$ be an odd integer then:

$\left({-x}\right)^n = -\left({x^n}\right)$

Proof

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

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

But $x^{2k} \ge 0$ from Even Powers are Positive.

The result follows.

Proof of Corollary

We have $\left|{-x}\right|^n = \left|-{\left({x^n}\right)}\right|$ and the result follows from the above.

$\blacksquare$

Sources

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