How to resolve a power of a negative number? $\left(-64\right)^{\left(\frac{3}{2}\right)}$
(Disclaimer - I work in a HS math center, helping students. This is from an Algebra/Trig text used by both sophomores and juniors depending on the class. Every so often, I hit an odd situation that I don't recall how to solve. Is this a paradox, or is there a preferred answer?)
I read thru How do you compute negative numbers to fractional powers? twice, and the issue isn't with i, the students know how to handle a simple square root of -1, it's more practical - do I suggest they raise to the power first, take the root first, or object to a non-integer power of a negative number? Much of the answer linked is well beyond their level. 

Personally I would square root first because the students know how to do this, giving $(8i)^3=-512i$.
As with all square root questions there are two possible answers - if you cube first and then take the square root you get $512i$. This suggests that there is a hidden convention at work - indeed there is, and it is a more subtle thing when complex numbers are involved.
Personally, I don't see the point of asking a question like this, which really belongs with Argand Diagrams and the modulus/argument form of complex numbers at the elementary level. It skates over too many issues which are liable to confuse without engaging the mathematical equipment or language for discussing them.
