Correct Interpretation of Simple Expression ?

[uart]

$$(3p^2 q)^3 \div 9 p q^2$$

The above is a simple algebra expression. The question concerns the correct interpretation of the divisor in the above example. Strictly speaking I would have thought that only “9” is the divisor here and if you wanted the whole “9pq^2” as the divisor then you should put it in parentheses. But it appears many people interpret the whole “9pq^2” as the divisor even without parentheses.

Strictly speaking, which is correct.

 

[Gokul43201]

When I saw this, I understood the denominator as 9pq^2. If I wanted only 9 in the denominator, I would use brackets to avoid confusion.

Strictly speaking, I think the convention is to group terms that are separated by $$\div$$ and then group terms separatied by *. If this is correct, then only the 9 belongs in the denominator. However, in this example the use of * is elliptic (implied, 9*p*q*q) and hence causes more confusion, and gives more room for misinterpretation.