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.

 

[tacman]

However, in mathematics, the concept of order of operations dictates that we first simplify the expression within the parentheses before dividing by the number outside of the parentheses. Therefore, both interpretations are correct and will result in the same answer. It is important to note that when simplifying expressions, it is best to use parentheses to avoid any confusion and ensure clarity in the final answer.