Domain of x^y ?

Hello,

My name's Eliott, I'm a student from Switzerland.

I haven't found on internet the domain of any number raised to any number so I tried to make it, please tell me if it's correct !

Justification :

I have done this domain by assembling three domains, each one corresponds to an interval of x. For each value of x we have the value of y corresponding for x^y to be defined.

All of the three domains are cartesian because there are two variables (A × B where A is the domain of x and B is the domain of y).

First case : if x is strictly negative, y has to be... This thing. First, it means that y has to be a rational number, negative powers of irrational numbers aren't defined. Also, y hasn't to be the quotient of an odd number over an even number. If it was the case, we would have x^(a/b) = b-th root of x^a. If a is odd, and x is strictly negative, x^a would also be strictly negative. And we know that an even n-th root of a negative number isn't defined. So if x is strictly negative and y belongs to Q but isn't an odd integer over an even integer, x^y is defined.

Second case : if x=0, y has to be strictly positive. 0^y isn't defined when y is negative or = 0.

Third case : if x is strictly positive, y belongs to R. So if x is strictly positive, there is no restriction for y. x^y is always defined.

I hope you liked this domain, I think that it's useful to know in every case if a power is defined. Now you are able to make it.

See you soon !