<<< posted as an answer as it nearly solves the question, and is too long for a comment >>>
@mikekaganski in tdf#69293:
‘And it’s not correct to spaculate that “(-8)^(1/3)=(-8)^(2/6)=((-8)^2))^(1/6)=64^(1/6)=2”. This speculation is a sort of sophism that masks the fact that there are more than one nth roots. When you make any transformations of powers you must take into account the possible loss of some roots or introduction of new roots. The correct result of “sqrt(x)” should be a multitude of two (possibly complex) numbers, not a single number. As we cannot return such a result in a spreadsheet we agree to see only positive answer, but it doesn’t mean that we don’t deserve to get correct negative real number as a cube root of a negative number.’
(i now remember that i did! read that in the past, just didn’t find it, neither remembered exact content),
it doesn’t teach which concept forbids extend or shortening of fractional exponents, but neatly explains why the results vary … at Mike,
would be nice if somebody can point to the math rules about it …
and as a ‘calc question’ it’s left if it’s ‘more correct’ that calc delivers -2 for -22/6 when it’s constructed by itself (and not by expanding the exponent of -21/3), (as calc does and is correct for a ‘shortened’ exponent), or if +2 would be better as users would expect acc. manual calculation?
mathematically one can justify both solutions? ‘-2’ holds for a re-conversion ‘second root of -2 to the power of 6’, just like ‘2’ doe’s, but slides down to the ‘odd roots’ while the original term formulates an ‘even root’?
‘parentheses dictate’ isn’t applicable imho because the way of calculation prescribes the resolution of the parentheses according to mathematical calculation rules,
in other words:
if ‘This speculation is a sort of sophism that masks the fact that there are more than one nth roots. When you make any transformations of powers you must take into account the possible loss of some roots or introduction of new roots.’ ‘forbids’ the expansion of 1/3 to 2/6, the conversion of 2/6 to 1/3 should also be avoided for fractional exponents, and thus +2 would be the correct result for 2/6 as exponent?
or in other other words:
-2 violates ‘As we cannot return such a result in a spreadsheet we agree to see only positive answer’, the specified exception refers to ‘odd roots’,
still a little confused …
P.S. got it?
squareroot and 6-th root are ‘even roots’, and thus have two solutions, +/-x, thus calculating from 2/6 you get directly to ‘also -2’, and the reduced exponent 1/3 just reduces the scope for solutions, but it’s result is correct,
question left: is it ok / legal to expand the exponent from 1/3 to 2/6?