Writers on algebra have noticed that the usual laws of indices lead to self contradictions (Chrystal called them “paradoxes”), no matter whether the meaning of a fractional index is defined by

or by

but it does not appear that anyone has yet shown how the laws can be modified in a way that makes them self-consistent. In this note, the necessary amendments are stated for each of these definitions, and are shown to be very much simpler for definition (1). This is fortunate; for that definition appears to be the generally accepted one, though definition (2) is adopted by Hardy and some others.