Sign of Odd Power

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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:

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


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