Simple property of power of -1

Question

Is $(-1)^{a+b} = (-1)^{a-b}$ true $\forall a,b \in \mathbb{Z}$ ?

My argument: $$ (-1)^{a-b} = \frac{(-1)^a}{(-1)^b} = \frac{(-1)^a}{(-1)^b}.\frac{(-1)^b}{(-1)^b} = \frac{(-1)^{a+b}}{(-1)^{2b}} = \frac{(-1)^{a+b}}{1} = (-1)^{a+b} $$

Any flaw?

Answer 1

Nope, no flaw. (apart from your lack of parentheses, that's confusing). We know that $(-1)^k$ is $1$ when $k$ is even, and $-1$ when $k$ is odd. And it's trivial that $a-b\equiv a+b\mod 2$.