An easy way to remember this is that a square with side m has an area equal to m times m, or m^2 (m-squared), and a cube with side m has volume equal to m times m times m, or m^3 (m-cubed). If m is the unit "meter," then you read these as copyright stated.
(m^2 means m with an exponent (superscript) of 2.)
Panj, Yes there really is Santa Clause .. and a 2^0.. anything raised to the 0 power is 1. It comes up lot in applied math.
Like English, math requires punctuation (especially without fonts or parentheses). 2^2/3 is either 2 to the two-thirds or 2 squared divided by three. Without explicit instructions it's 2 squared divided by three but I am not sure that's what you intended.
One of many factors that influences how people say these things is familiarity. For some, these little superscripts were a topic of passing interest, and little interest at that. They remember the words and expressions used when they were taught about the concept.
For others, they became a topic of everyday conversation - an element of routine working activity. The wordy expressions used at first have been overwhelmed by the forms used in routine and rapid communication. In any case, those wordy expressions don't work when the exponent is something long and complex.
Hence etb's "a to the b" and my "X to the Y".