Given a rational number, would it be possible to know whether the root or some other power of the number is an irrational number? Can an automaton be designed for such a purpose?
An irrational number is an infinite string, and if you want an automaton that can read it, it will need to continue reading infinitely.
You cannot build a decider (a machine that always halts with output true or false), but you can build an acceptor (a machine that halts with false, but continues forever for true), which is what I believe you're asking.
Consider a machine that accepts the irrational number of the form
0.10110111011110111110...
Where the lengths of runs of 1s is always growing between 0s. It's relatively easy to define a Turing machine that can accept this number.
(For the implementation for such a machine, I'd suggest The Annotated Turing, which also has an implementation for a machine that accepts √2.)