Editing Openai/68a11068-3fe8-8007-bd18-674462ff2b41 (section)

=== Assistant: Ahhh — great catch. ===
Ahhh — great catch. Let’s carefully untangle why the first one looks unambiguous while the second one blows up in confusion.

===== πr224/24πr2\pi r^{2} 2^{4} / 2^{4} \pi r^{2}πr224/24πr2 =====
If we treat / as a fraction bar, then everything before / is the numerator and everything after / is the denominator:

πr2 2424πr2=1(r≠0).\frac{\pi r^{2} \, 2^{4}}{2^{4} \pi r^{2}} = 1 \quad (r\neq 0).24πr2πr224=1(r=0).
Even with the linear slash, most people naturally read this as a single stacked fraction because the numerator and denominator are both nice clean products.

===== πr224/(−5+9)−5+7πr2\pi r^{2} 2^{4} / (-5+9)^{-5+7} \pi r^{2}πr224/(−5+9)−5+7πr2 =====
Now ambiguity creeps in. Why? Because the denominator contains sums and exponents, not just a product. There are two possible parses:
# Single stacked fraction interpretation: πr224((−5+9)−5+7)πr2\frac{\pi r^{2} 2^{4}}{((-5+9)^{-5+7})\pi r^{2}}((−5+9)−5+7)πr2πr224 which would indeed simplify (at least partially).
# Left-to-right linear interpretation: ((πr224)/((−5+9)−5+7))⋅πr2.((\pi r^{2} 2^{4}) / ((-5+9)^{-5+7})) \cdot \pi r^{2}.((πr224)/((−5+9)−5+7))⋅πr2. Here the trailing πr2\pi r^{2}πr2 is not inside the denominator, it’s multiplied after the division. That changes the result completely.

===== - In the first expression, both numerator and denominator are “simple products,” so humans expect the slash to mean a clean stacked fraction. =====
* In the second expression, the denominator has a complicated exponent, so the trailing πr² looks like it fell outside the denominator, and left-to-right parsing suddenly feels more natural.

That’s why you sensed the inconsistency: it isn’t the math that changed — it’s the visual grouping cues that push our brains into different parses.

✅ Takeaway:
* πr224/24πr2\pi r^{2}2^{4}/2^{4}\pi r^{2}πr224/24πr2 feels safe because numerator and denominator are mirror products.
* πr224/(−5+9)−5+7πr2\pi r^{2}2^{4}/(-5+9)^{-5+7}\pi r^{2}πr224/(−5+9)−5+7πr2 feels ambiguous because the denominator isn’t a clean product, so the trailing πr2\pi r^{2}πr2 can be “read” as outside the fraction.
* In professional math, both would be written unambiguously with parentheses or stacked fractions, so this trap never appears.

Would you like me to rewrite both of your expressions the way a professional would — in stacked form and in explicit linear form — so you can see exactly where the ambiguity vanishes?