Question

Can an irrational number be raised to a non-trivial rational power to result in a rational value?

Trivial powers include zero, and things like multiples of 2 for sqrt(2).

Answer 1

\[ \bf No\quad it \quad can't.\]
And I can prove it by considering the opposite assumption

Answer 2

Basically irrational number is the infinite sequence of digits without stabilization on permanent repetition of the same "word".

Answer 3

Rational , on the opposite, stabilizes on the same word. Suppose that the result's digits stabilize on the same word. This repeting word -being multiplied by smaller and smaller and ad-infinitum smaller powers of 10 is depending more sensitively and more sensitively on the appropriate finite subsequence of the irrational base. The unrepeatable property of that INPUT (digits of base) prevents repeatable OUTPUT with the repeatable digits of the other input: digits of the power.