Difficulty understanding division sign in expression

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.

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.

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.

Short answer:

"Should I interpret the division sign as follows:$\left(\frac{64x^3}{27a^{-3}}\right)^\frac{-2}{3}$"

Yes. That is the correct interpretation.

"Or as I originally interpreted it: $\left[\left(\frac{64x^3}{27}\right)\times a^{-3}\right]^\frac{-2}{3}$"

No. But that was a perfectly reasonable (unfortunately incorrect) interpretation.

Long (and weird) answer:

A statement $A \div B$ implies that we think of $B$ as a single "chunk" and it is conventional that $27a^{-3}$, if it is expressed is a single "thing". Why? Hmmm, it's a good question.

This probably isn't a good answer but I think of it this way: Multiplication takes precedence over addition (that is because we can distribute multiplication over addition: $a(b+c) = (b*c) + (b*c)$) and makes addition, in my mind, a fluid ongoing modification, whereas multiplication is a "gluing" bonding modification. We never consider $a - b + c$ to mean $a - (b+c)$ but always to mean $(a-b) + c$ because there is nothing "permanently bonding" about $b+c$ so that whenever we see $b+c$ we think "wow, that is solid 'chunk'".

Multiplication however "feels" different. It is a bonding. In my intuition, it feels almost "chemical" in nature. Where as $a + b - c + d +f$ seems like a liquid flowing process, $27a^{-3}$ seems like a crystalline little rock pebble.

In a way this is why prime factorization is so critical whereas sums are not. If you want to solve $n + m = 27; n,m \in \mathbb N$ we can just let $n$ be anything from $1$ to $26$ because addition is "fluid" and we can break it anywhere. But if you want so solve $n*m = 27; n, m \in \mathbb N$ it is far more finicky. Not any old natural number will divide into $27$. You must tap at it until you find a "subparticle" such as $3$ or $9$ and chisel out the rest.

(Lots of rock and liquid metaphors.)

But... obviously, if I got before a math class and tried to teach that way, my students would ... be perplexed. That's not math, that's ... impressionism.

Well, remember the rules. There's some dumb mnemonic about the orders of operation all you young kids are using these days[$*$]. I can never remember it but ... Let multiplication take precedence over the division symbol. Just... obey, and stop asking questions.... I guess.

Anyhow, unfortunately for you $A\div B$ is really ambiguous. Fortunately "serious" mathematicians stop using it precisely for that reason and pretty soon you'll just use fraction notation $\frac AB$ and won't ever have to worry about this again.

[$*$] I was thinking of PE(M/D)(A/S) which WOULD indicate $5\div 2\times 3$ is $(5\div 2)\times 3$ and not $5\div (2\times 3)$. So basically according to that $64x^3 \div 27a^{-3}$ would be $(64x^2*\frac 1{27} a^{-3})$. But the mnemonic is, apparently wrong. I'd modify is as PE(Coeffecient terms)(M/D)(A/S) so that $5\div 2\times 3 = \frac 52\times 3$ whereas $5\div 2a^2$ is $\frac 5{2a^2}$. However what is $5\div ab$? is $ab$ a coefficient term? Hmmm, I think my assumption would be that it is. Or, at least that'd be what I'd assume if I didn't consciously think about it specifically. But it is ambiguous.

$(64x^3 \div 27a^{-3})^{\frac{-2}{3}}$ should be interpreted as $(64x^3 \div 27 \cdot a^{-3})^{\frac{-2}{3}}$. Using the order of operations rules, multiplication and division have the same operator precedence, so they are evaluated from left to right. A similar question went viral a while back. The question was: "What is $6 \div 2(1+2)$". At first sight, this might seem to be $6 \div (2 \cdot 1 + 2 \cdot 2)=6 \div (6) = 1$, but in reality it should be $(3)(1+2)=3(3)=9$.