Fraction of a Fraction order of operation: $\pi/2/\pi^2$

Question

I think I'm having a bit of a senior moment or something... I was evaluating $\pi/2/\pi^2$. My instinct was that the answer would be $\frac{\pi^3}2$ from $\frac{\pi}1 * \frac{\pi^2}2$ but I checked wolfram and got $\frac{1}{2\pi}$.

I'm pretty sure I understand how wolfram evaluated and got that result, but now I have the questions: which is correct, and why?

Symbolab gave me the answer I first assumed: $\frac{\pi^3}2$ with the same input as wolfram and I'm pretty sure they can't both be correct.

The links to the equations I've used are:

https://www.symbolab.com/solver/step-by-step/%5Cfrac%7Bpi%7D%7B%5Cfrac%7B2%7D%7Bpi%5E%7B2%7D%7D%7D

https://www.wolframalpha.com/input/?i=pi%2F2%2Fpi%5E2

The question I was trying to solve when I came across this was $\int_{1/2}^1 x \cos(\pi x) dx$

Thanks for your assistance!

edit: Thanks for the replies so far, I think what I'm trying to get a feel for is how to know around which terms I should use brackets when I see this type of fraction.

Answer 1

These are two distinct inputs. $\pi / 2 / \pi^2$ (wolfram input) is interpreted as $\frac{\frac{\pi}{2}}{\pi^2} = \frac{\pi}{2 \pi^2}$ because of the left-to-right order of performing operations, while if you introduce parentheses as in $(\pi) / (2 / (\pi^2))$ (symbolab input) it's interpreted as $\frac{\pi}{\frac{2}{\pi^2}} = \frac{\pi^3}{2}$, which is why you get two different results.

Answer 2

Wolfram and all computer languages I know will evaluate $a/b/c$ from left to right, giving $a/b/c=(a/b)/c=a/(bc)$ This is a good reason to use parentheses in this expression to make sure you get what you want.