a lot of power

check the last digit if it is even answer is 1 or else -1

$(-1)^{2n \pm 1} = -1$ and $(-1)^{2n} = 1$ where $n$ is a positive non-zero integer

So can we write $(-1)^{99999994378}$ as described in the way above

If $99999994378 = 2n \pm 1$ Then

$99999994378 \mp 1= 2n$

So

$99999994377 = 2n \Rightarrow n = 49999997188.5$

and

$99999994379 = 2n \Rightarrow n = 49999997189.5$

Neither of those answers fit the description of $n$ so that means that

$99999994378 = 2n$

And since

$(-1)^{2n} = 1$

We can say that

$(-1)^{99999994378} = 1$

$(-1)^{2n}=1$ that,s it.