What's the intuition behind non-integer exponents/powers Consider some $a \in \mathbb{R}$ and $x \in \mathbb{R}\backslash \mathbb{N}$. 
Is there some intuition to be had for the number $a^x$?
For example the intuition of $a^2$ is obvious; it's $a*a$ which I can think about with real world objects such as apples (when $a \in \mathbb{N}$). What about $a^{1.9}$?

If we fix $a>0$, $f(x)=a^x$ is continuous on $\mathbb R$. The intuition behind rational exponents is pretty clear, and one extends from the rationals to all reals in this manner.
