Did you eventually realise that the expression a^b should be understood to "really" mean exp(b * ln(a)), at least in the case that b might not be an integer?

I think even in complex analysis, the above definition a^b := exp(b ln(a)) makes sense, since the function ln() admits a Riemann surface as its natural domain and the usual complex numbers as its codomain.

[EDIT] Addressing your response:

> Calculus glosses over the case when a is negative

The Riemann surface approach mostly rescues this. When "a" is negative, and b is 1/3 (for instance), choose "a" = (r, theta) = (|a|, 3 pi). This gives ln(a) = ln |a| + i (3 pi). Then a^b = exp((|a| + i 3 pi) / 3) = exp(ln |a|/3 + i pi) = -|a|^(1/3), as desired.

Notice though that I chose to represent "a" using theta=3pi, instead of let's say 5pi.

I see what GP's point is, high-school-level calculus generally restricts itself to real numbers, where the logarithm is simply left undefined for nonpositive arguments. After all, complex analysis has much baggage of its own, and you want to have a solid understanding of real limits, derivatives, integrals, etc. before you start looking into limits along paths and other such concepts.

Even then, general logarithms become messy. It's easy to say "just take local segments of the whole surface" in the abstract, but any calculator will have to make some choice of branch cuts. E.g., clearly (−1)^(1/3) = −1 for any sane version of exponentiation on the reals, but many calculators will spit out the equivalent of (−1)^(1/3) = −e^(4πi/3) instead.

(Just in general, analytic continuation only makes sense in the abstract realm. If you try doing it numerically to extend a series definition, you'll quickly find out how mind-bogglingly unstable it is. I think there was one paper that showed you need an exponential number of terms and exponentially many bits of accuracy w.r.t. the number of steps. Not even "it's 2025, we can crank it out to a billion bits" can save you from that.)

The problem is a^b := exp(b ln(a)) sort of breaks down when a is negative, which is a case that is covered in algebra class but glossed over in calculus.