It's not 1 that is exactly the problem. It is also 0. And perhaps - 1 although that is a lot trickier than the other two.
Let's start with one. It give 1^1 = 1 for a power of 1. It gives 1^2 = 1 * 1 for a power of 2. It will give 1^100000 = 1 for a power of 100000.
So the base can be 1 but it is known that any power won't change it's value.
The same can be said of 0. 0^1000 = 0^1 = 0. 0^0 is really a bad dude. You won't learn about that for awhile.
(-1)^11 is different from (-1)^10. The first gives -1 and the second gives 1. So you have to be careful with -1.
There is nothing wrong with (1/4)^3. The power of three and the base of 1/4 are fine. The answer is (1/4)*(1/4) * (1/4) = 1/64 which is perfectly good answer.
It works for negatives too. (-1/2)^3 = - 1/8