I Error propagation of exponentials

[TheCanadian]

I am just wondering why there is a discrepancy between two different methods for error propagation. For example, if you have $Q = (a)(b)(c)$ then the relative error in Q is simply the square root of the sum of the squares of each of the terms being multiplied together, correct? But what if $Q = (a)(a)(a)$. Why isn't the relative error in Q now simply once again the square root of the sum of the squares of a (which in this case would be 3 terms)? I understand the derivation for the relative error in $Q = a^3$ being $3 \frac {\Delta a}{a}$ but just don't quite understand why the earlier rule pertaining to basic multiplication and division no longer applies. What is the reason for a discrepancy between the two methods of error propagation? Can't exponentiation (using positive integers) be considered as just an extension of multiplication?

 

[Knower]

Notice that in Q = (a)(b)(c) the terms a, b and c are considered to be independent in the sense that the error of one of them is independent from the error to another. But if a = b = c then the "three" terms are not independent.

 

[Charles Link]

One suggestion is to consider a function like S=(A+B)^2 where A and B are two variables that can each take on the values of +1 and -1. If A and B are uncorrelated, you will have less spread in the S distribution than if A=B. In computing experimental uncertainties, often the "delta" is just an estimate and it can be difficult to account for correlations in the variables that are often considered to have random uncertainties.