What is the domain of this function? (I am so confused)

Hello.

I would like to ask what is the domain of the following fucnction.

y=(5x^x)^(x+1)

Our teachers tell us that the domain of a power function (A^X where A is constant and X is a variable) is that the base must be positive and must differ from 1,so in the above example, it is logically that the solution of the following system of inequalities defines the domain:

5x>0 and

5x<>1 and

5x^x>0 and

5x^x<>1

Am I right? And I must note that in our textbook the power function's domain(sorry if I misuse mathematical argo) A^X is that the base A must be positive but it does not say that it must differ from 1. Ohhhh! Please clarify this very obscure situation!

Must a base differ from 1?

And what if we take X for 1 so we would get number 1 in the base ,which is prohibited. So probably the domain must be x>0 and x<>1 ?

Thanks for replying