Can anyone please tell me how to define $x^\sqrt 2$ ?
If it was $x^2$ or $x ^ \frac{1}{2}$, we could have said that $x^2$ means $x \times x$ and $x ^ \frac{1}{2}$ means a number y such that $y^2 = x$.
But how to define $x^\sqrt 2$ ?
Can anyone please help me ?
Take a sequence of rationals $q_n$ converging to $\sqrt 2$. Then $x^{\sqrt 2}$ is defined as the limit of the sequence $x^{q_n}$. From continuity of the exponent, this limit exists, is unique, and independent of the choice of sequence. (And as you said, you already have a definition for each term in the sequence, since the powers are rational)