Answer
$\begin{split}
(-1)^n & = (-1)(-1)(-1)(-1)...(-1) \\
& = (-1)^2(-1)^2...(-1) \\
& = 1(-1) \\
& = -1 \\
\end{split}$
Work Step by Step
Let $n$ be an odd integer.
$a$ can't be equal to zero, because zero is even $(0 = 2.0)$
If a \geqslant 0:
$(-1)^n = (-1)^1(-1)^1(-1)^1(-1)^1...(-1)$
$n$ times, but $n$ is odd, so we can group the pairs:
$(-1)^n = (-1)^2(-1)^2...(-1)$
and we'll have 1 left. Knowing that $(-1)^2 = 1$, we can do:
$\begin{split}
(-1)^n & = 1.1.1.1...(-1) \\
& = -1 \\
\end{split}$
If a < 0 (similar):
$(-1)^n = (-1)^{-1}(-1)^{-1}(-1)^{-1}(-1)^{-1}...(-1)^{-1}$
But $(-1)^{-1} = \dfrac{-1}{1} = -1 = (-1)^1$, now repeat the same process we did when a > 0.
So $(-1)^n$ (with $n$ being and odd number), is always equals to -1.
