It's a bit more involved than that.

  mul rcx, rax       ; multiply our result by our base, save into rcx

This should probably be using rbx, since that's the designated base, not rax, which is the exponent.

In the explanation just above, there's a 2*8 step missing, while this one is incorrect:

  4. result = 16 * 2, result is now 16

Since 2*16 is not 16.