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

== gpt-5-thinking ==
=== User: 50/21+4 ===
50/2(1+4)

=== Assistant: It’s ambiguous as written. ===
It’s ambiguous as written.

If you follow standard conventions (× and ÷ have equal precedence; evaluate left→right):
1+4 = 5 → 50/25 → 255 = 125.

If you intended the denominator to be 2(1+4):
50 ÷ [2(1+4)] = 50 ÷ 10 = 5.

Write it clearly as either (50/2)(1+4) (gives 125) or 50/(2(1+4)) (gives 5).

=== User: solve πr²2⁴/2⁴πr² ===
solve πr²2⁴/2⁴πr²

=== Assistant: πr2 2424 πr216 πr216 πr21for r≠0.\frac{\pi r^{2}\,2^{4}}{2^{4}\,\pi r^{2}} ===
πr2 2424 πr2=16 πr216 πr2=1(for r≠0).\frac{\pi r^{2}\,2^{4}}{2^{4}\,\pi r^{2}}
=\frac{16\,\pi r^{2}}{16\,\pi r^{2}}
=1\quad(\text{for }r\ne 0).24πr2πr224=16πr216πr2=1(for r=0).
Note: the original is undefined at r=0r=0r=0 since it would be 0/00/00/0.

=== User: if 50/21+4  125 then what does πr²2⁴/2⁴πr² ===
if 50/2(1+4) = 125 then what does πr²2⁴/2⁴πr²

=== Assistant: Under the same left-to-right rule that makes 50/21+412550/21+412550/21+4125, ===
Under the same left-to-right rule that makes 50/2(1+4)=12550/2(1+4)=12550/2(1+4)=125,

πr224/24πr2    =    ((πr2⋅24)/24)⋅πr2    =    πr2⋅πr2    =    π2r4.\pi r^{2}2^{4}/2^{4}\pi r^{2}
\;\;=\;\;((\pi r^{2}\cdot 2^{4})/2^{4})\cdot \pi r^{2}
\;\;=\;\;\pi r^{2}\cdot \pi r^{2}
\;\;=\;\;\pi^{2}r^{4}.πr224/24πr2=((πr2⋅24)/24)⋅πr2=πr2⋅πr2=π2r4.
(If you instead mean a single stacked fraction πr22424πr2\frac{\pi r^{2}2^{4}}{2^{4}\pi r^{2}}24πr2πr224, then it simplifies to 111 for r≠0r\neq 0r=0. The difference is all about parentheses/formatting.)

=== User: if πr²2⁴/2⁴πr²  1 then what is 50/21+4 equal to? ===
if πr²2⁴/2⁴πr² = 1 then what is 50/2(1+4) equal to?

=== Assistant: 5. ===
# 
Reason: If you interpret πr224/24πr2\pi r^22^4/2^4\pi r^2πr224/24πr2 as one stacked fraction (so it simplifies to 111), you’re grouping the entire 24πr22^4\pi r^224πr2 in the denominator. By the same convention,

50/2(1+4)=502(1+4)=502⋅5=5010=5.50/2(1+4)=\frac{50}{2(1+4)}=\frac{50}{2\cdot5}=\frac{50}{10}=5.50/2(1+4)=2(1+4)50=2⋅550=1050=5.

=== User: when you no evaluate πr²2⁴/2⁴πr² as a stacked fraction ===
when you no evaluate πr²2⁴/2⁴πr² as a stacked fraction

=== Assistant: Then you treat it left-to-right since × and ÷ have equal precedence: ===
Then you treat it left-to-right (since × and ÷ have equal precedence):

πr224/24πr2=((πr2⋅24)/24)⋅πr2=πr2⋅πr2=π2r4.\pi r^2 2^4/2^4 \pi r^2
= \big((\pi r^2\cdot 2^4)/2^4\big)\cdot \pi r^2
= \pi r^2 \cdot \pi r^2
= \pi^{2} r^{4}.πr224/24πr2=((πr2⋅24)/24)⋅πr2=πr2⋅πr2=π2r4.
(Defined for all rrr; no stacked fraction grouping.)

=== User: is πr²2⁴/2⁴πr²  πr²2⁴/πr²2⁴ ? ===
is πr²2⁴/2⁴πr² = πr²2⁴/πr²2⁴ ?

=== Assistant: Short answer: it depends how you’re grouping. ===
Short answer: it depends how you’re grouping.
* Left-to-right (no stacked fraction): πr224/24πr2=π2r4,πr224/πr224=28=256.\pi r^{2}2^{4}/2^{4}\pi r^{2}=\pi^{2}r^{4},\qquad \pi r^{2}2^{4}/\pi r^{2}2^{4}=2^{8}=256.πr224/24πr2=π2r4,πr224/πr224=28=256. So they’re not equal in general (only equal if π2r4=256 \pi^{2}r^{4}=256π2r4=256).
* As single stacked fractions: πr22424πr2=1,πr224πr224=1,\frac{\pi r^{2}2^{4}}{2^{4}\pi r^{2}}=1,\qquad \frac{\pi r^{2}2^{4}}{\pi r^{2}2^{4}}=1,24πr2πr224=1,πr224πr224=1, so they are equal (both 1).

=== User: -5+9⁻⁵⁺⁷  2⁴. ===
(-5+9)⁻⁵⁺⁷ = 2⁴. It that is true, then πr²2⁴/(-5+9)⁻⁵⁺⁷πr² = 1 and πr²2⁴/2⁴πr² =1

=== Assistant: You’re right—if the exponent is the whole −5+7-5+7−5+7, then ===
You’re right—if the exponent is the whole (−5+7)(-5+7)(−5+7), then

(−5+9)−5+7=4 2=16=24.(-5+9)^{-5+7}=4^{\,2}=16=2^{4}.(−5+9)−5+7=42=16=24.
So, when read as single stacked fractions,

πr224(−5+9)−5+7 πr2=πr22424πr2=1(r≠0).\frac{\pi r^{2}2^{4}}{(-5+9)^{-5+7}\,\pi r^{2}}
=\frac{\pi r^{2}2^{4}}{2^{4}\pi r^{2}}
=1\qquad (r\neq 0).(−5+9)−5+7πr2πr224=24πr2πr224=1(r=0).
The only time it wouldn’t be 111 is if the linear slash makes the trailing πr2\pi r^{2}πr2 sit outside the denominator, i.e.

πr224/(−5+9)−5+7 πr2=(πr22424)πr2=π2r4.\pi r^{2}2^{4}/(-5+9)^{-5+7}\,\pi r^{2}
=\Big(\frac{\pi r^{2}2^{4}}{2^{4}}\Big)\pi r^{2}
=\pi^{2}r^{4}.πr224/(−5+9)−5+7πr2=(24πr224)πr2=π2r4.
So the math agrees with you; the discrepancy is purely about parsing. Write it as a stacked fraction or with parentheses to lock in the meaning.

=== User: But you are contradicting and ambiguous. ===
But you are contradicting and ambiguous.  If πr²2⁴/2⁴πr²  is treated as a stacked fraction, but it's equivalent expression πr²2⁴/(-5+9)⁻⁵⁺⁷πr² is not.  Both expression are stacked due to implicit mulitiplication

=== Assistant: You’re right—I was inconsistent. ===
You’re right—I was inconsistent. Let’s lock one convention and stick to it.

===== Convention (professional math style) =====

Treat the slash as a fraction bar that spans the entire following product (juxtaposition).
So a/bca/bca/bc means a/(bc)a/(bc)a/(bc), not (a/b)c(a/b)c(a/b)c.

Under this rule:

πr22424πr2=1(r≠0),\frac{\pi r^{2}2^{4}}{2^{4}\pi r^{2}}=1 \quad (r\ne 0),24πr2πr224=1(r=0),
and since (−5+9)−5+7=42=16=24(-5+9)^{-5+7}=4^{2}=16=2^{4}(−5+9)−5+7=42=16=24,

πr224(−5+9)−5+7 πr2=πr22424πr2=1(r≠0).\frac{\pi r^{2}2^{4}}{(-5+9)^{-5+7}\,\pi r^{2}}
=\frac{\pi r^{2}2^{4}}{2^{4}\pi r^{2}}
=1 \quad (r\ne 0).(−5+9)−5+7πr2πr224=24πr2πr224=1(r=0).
So both expressions equal 1 with that (professional) parsing.
My earlier “not 1” came from switching to the calculator/programming left-to-right rule midstream. I’ll stick to the professional convention going forward.