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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?

  • You might want to start by reading It ain't no repeated addition (and the follow-ups). Basically, while the multiplication of two natural numbers can be seen as repeated addition, this is not really the right mathematical definition. Similarly, while exponentiation can be seen as repeated multiplication, this isn't really right. More typically, exponentiation is defined via a Taylor series expansion or via a differential equation. – Xander Henderson Jul 22 '18 at 23:28
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    Duplicate of What Is Exponentiation?. See also What does $2^x$ really mean when $x$ is not an integer? – dxiv Jul 22 '18 at 23:29
  • "how can we understand a power of a[n] irrational number $x^r$?" Perhaps you meant to ask how we understand an irrational power of a (positive real) number $x$. In any case one can define it by an approximating sequence. – hardmath Jul 22 '18 at 23:30
  • @dxiv I was looking more for an explaination for real powers, so I tought that "what is exponentiation" is for all powers including complex powers. but a friend of mine added another answer . So now it adresses my question good enough... –  Jul 29 '18 at 21:24

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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.

mweiss
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