when a complex power is single-valued

Since z^a = e^(a log z) inherits the multi-valuedness of the logarithm, it is natural to ask when, if ever, the ambiguity goes away on its own, so that z^a is a single number without any branch choice. The answer depends entirely on the exponent a, and there is a clean rule.

Different choices of log z differ by 2 pi i k for integer k, so different values of z^a = e^(a log z) differ by the factor e^(a (2 pi i k)) = e^(2 pi i a k). The power is single-valued for all z exactly when this factor equals 1 for every integer k, which happens precisely when a is an integer. If a is a rational number p/q in lowest terms, the factor e^(2 pi i (p/q) k) cycles through exactly q distinct values, so z^(p/q) is finitely multi-valued with q values, the q-th roots situation. If a is irrational or genuinely complex (like a = i), the factor e^(2 pi i a k) takes infinitely many distinct values, so z^a is infinitely multi-valued.

So the hierarchy is: integer exponent gives one value (the ordinary power); rational exponent p/q gives q values (roots); any other exponent gives infinitely many. This is why z^2 and z^(-3) are honest single-valued functions you can manipulate freely, while sqrt(z) has two values and z^i has infinitely many. The practical upshot: only for integer exponents may you treat z^a as a plain single-valued function on the whole punctured plane; otherwise you must commit to a branch (usually the principal one) and accept a cut.