TITLE: Difficulty understanding division sign in expression
QUESTION [13 upvotes]: 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

REPLY [8 votes]: 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.