Q:

Difficulty understanding division sign in expression

I have difficulty understanding following expression:
$$(64x^3÷27a^{-3})^\frac{-2}{3}$$
Should I interpret the division sign as follows:
$$\left(\frac{64x^3}{27a^{-3}}\right)^\frac{-2}{3}$$
Or as I originally interpreted it:
$$\left[\left(\frac{64x^3}{27}\right)\times a^{-3}\right]^\frac{-2}{3}$$
The reason for my confusion is the order of operations as we can write a division as a multiplication (which is how I came to my original interpretation):
$$\left[64x^3\times\frac{1}{27}\times a^{-3}\right]^\frac{-2}{3}$$
I know this is wrong as I cannot get the correct answer this way. This leaves me with the conclusion that the division sign they use is the same as everything after it should be on the bottom of the fraction?
The link to the exercise including the solution:
Mathopolis exercise

A:

This is a good question. Without realizing my assumptions, I interpret $(64x^3 \div 27a^{-3})^\frac{-2}{3}$ as $(\frac{64x^3}{27a^{-3}})^\frac{-2}{3}$. I realize now there is an implicit pair of parentheses around the expression $27a^{-3}$. Clearly, the authors of that question assume parentheses around that term as well. 
It is always good to be aware of the assumptions you bring to a problem, so thank you for bringing this up. In general, however, I wouldn't expect to see a lot of problems with division signs like that. Most rational functions (in calculus, for example) are written as fractions; higher math seems to eschew the grade-school division sign.  

A:

The "right" answer, in the sense that it is the generally accepted convention, is that you are correct; multiplication and division are performed in a single pass left to right, so 
$$64x^3÷27a^{-3}$$
parses as
$$((64 x^3) \div 27 )a^{-3}$$
However, many (most?) people don't really learn the convention, and write what they think "looks" right. There are also some people that learn the convention wrongly, thinking that multiplication and division are to happen in separate passes.
Thus, it is unfortulately common for people to write such an expression when they actually mean for it to be parsed as
$$(64x^3)÷(27a^{-3})$$
So how should you interpret this expression? Unfortunately, there is no rule here: you have to guess what the author intended. Sometimes, the surrounding context (e.g. the previous or next step in a calculation) can give you clues as to which is meant.
And you should never write such expressions if you can avoid it, since it's prone to having the reader misunderstand.

A:

I have always interpreted coefficients and variables next to each other to have an order of operation higher than MD. This is because when someone is creating an expression or equation, and they intend for that $ \div $  or $ \times $ to be used with what looks like the coefficient, they will notice the ambiguity instantly and put the parenthesis in. This is my basis for the assumption.
However, there is no real set convention for dealing with the order of operations in such a scenario. It's good to state how you interpreted the question before starting, and if you can, try to derive what the calculation should be based on the context (or even the complexity) of the calculation.

