Powers of numbers behave in surprising ways...

Take a look at the following and try to explain what's going on.

Work out $2^1, 2^2, 2^3, 2^4, 2^5, 2^6$...

Work out $1^n + 2^n + 3^n + 4^n$ for some different values of $n$.

Here are some suggestions to start you off:

$4^n + 5^n + 6^n$

$2^n+3^n$ for odd values of $n$

$3^n + 8^n$

$2^n + 4^n + 6^n$

$3^n + 5^n + 7^n$

$3^n - 2^n$

$7^n + 5^n - 3^n$

$2^n+3^n$ for odd values of $n$

$3^n + 8^n$

$2^n + 4^n + 6^n$

$3^n + 5^n + 7^n$

$3^n - 2^n$

$7^n + 5^n - 3^n$

You may also like to take a look at Big Powers.

Click here for a poster of this problem.