Work out $9^n + 1^n$ for a few odd values of $n$.

What do you notice?

Can you prove it?

Now try the same with the following:

$7^n + 3^n$, where $n$ is odd.

$8^n - 2^n$, where $n$ is even.

$6^n - 4^n$, where $n$ is even.

Can you find any more results like these?