One can understand the concept of natural power, as $x^n$, being a product of a number by itself $n$ times: $$x^n=\underbrace{x\cdot x\cdot\dotsb\cdot x}_n$$
We can also get the idea of a rational power, $x^{\frac{p}{q}}=\sqrt[q]{x^p}$ , looking for the $q$-th root of a number, where $q$ is a integer, so it is the same idea as a power of a natural number.
But how can we understand a power of a irrational number $x^r$? Of course we can define it as the limit of the power of the rational number who is tending to the number $r$, but is there a better more intuitive way of explaining this, staying with the number $r$, instead of its approximation?
I don't know if you'd consider it "intuitive" or not, but the usual way to define an irrational power of a (positive) real number is via: $$a^r = \exp \left(r \log(a)\right)$$ where $\exp(u)$ denotes the exponential function $u \mapsto e^u$ and $\log$ denotes the logarithm base $e$.
Now, I hear you ask, isn't this circular, in that $e\mapsto e^u$ an example of precisely the kind of exponentiation the question is asking about? And the answer to that is: we can define $\log x$ as the integral $\int_1^x \frac{1}{t} dt$, and then define $\exp x$ as the inverse function of $\log x$. Neither of those definitions requires any restriction on what type of number $x$ is. Once $\log$ and $\exp$ are defined, $a^r$ can be defined as above.