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In exponentiation, we are told that raising something to an integral power (n, say) means multiplying it with itself a total of n times, if n is non-negative. And we also learn fairly early on that should the exponent be negative, then $a^{-n}=\frac{1}{a^n}$.

We even extend this to the rational field, where with $a^{\frac{p}{q}}$, we are led to believe that there is a unique number $s$, such that $s^q=a \implies s= a^{\frac{1}{q}}$, and we can hence define $a^{\frac{p}{q}}$ as $s^p$.

But what does it mean to have an irrational power? Clearly $2^{\pi}$ is not 2 multiplied by itself pi times...

Similarly, what does it mean to have an complex power, such as $2^{1+i}, 2^{i}$, and is there an explicit formula for calculating such numbers?

$\textbf{My thoughts}$

Going with my example, since we can approximate pi with rational numbers, e.g. 3.14, 3.14159, etc,I intuitively feel that it approaches a certain limit as we approach ever more accurate approximations of $\pi$- do we take that limit to be $2^{\pi}$?

Regarding imaginary exponents, I have heard of Euler's formula, $e^{ix}=\cos(x) +i\sin (x)$. Do we use this in calculating numbers with complex exponents?

e.g. $2^i=e^{\ln(2^i)}=e^{i\ln (2)}= \cos(\ln(2))+ i\sin(\ln (2))$?

However, what does it $\textbf{mean}$ to raise something to a complex power, and do the rules of exponentiation and logarithms still apply when dealing with them?

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