[mannerheim]

The radical symbol always refers to a single-valued function by convention.

x^(1/n) is used for multi-valued functions in complex analysis, not the 'n-radical' symbol, which would usually refer to the principal root.

[thehappypm]

Sure, but doesn’t that weaken the proof? It is an arbitrary restriction on the function that is required

[mannerheim]

Barely? It proves it for the principal root, and you can trivially prove it for the negative root using the principal root's irrationality.