114: Square Root Exponents

We all know that adding/subtracting exponents corresponds to multiplying/dividing the terms, like this:

$x^4 * x^7 = x^{11}$

$\dfrac{x^{14}}{x^9} = x^5$

Then negative exponents logically followed: $x^{-7} = \dfrac{x^2}{x^9}$

Then $\dfrac{x^3}{x^3} = x^0 = 1$ logically followed that. 

Additionally, multiplying/dividing the exponents relates to powers/roots

$ {x^4}^2 = x^{4*2} = x^8$

$\sqrt{x^6} = x^{6/2} = x^3$

So a fractional exponent means a radical, depending on the denominator of the exponent.

So here's my question:

What should we think about $x^{\sqrt{2}}$

How should we interpret that?