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 ?