help-19

[fluid tundra]

this question is garbage

[jade ice]

oh

[narrow crypt]

what

why am i pinged

0_0

[jade ice]

.

mqnic said dont use ai generated questions

i dont get it what is wrong wtih it

but i couldnt find it

i might actually listen to his idea this question doesnt makes sense...

the answer is garbage...

[fluid tundra]

there are math books full of questions, online repositories full of questions, full of them

pull up any one of those and get a question from there

[jade ice]

let me see if i can do few problems from it

[fluid tundra]

you can also ask AI for sources of questions depending on your needs, it can probably give you those

but whatever you do, do not ask it to generate a question itself. you will be wasting your time (and the time of others if you ask for help on these bad questions)

there are a lot of things AI is good for. generating questions is not one of those things

[jade ice]

https://cdn.discordapp.com/attachments/1395994787777216515/1404002999331852308/IMG_20250810_125506.jpg?ex=68999b30&is=689849b0&hm=099e49ffe4933c5a05eef632434e9db37336d81ed48dadeffedc6b2fef9c61d5&

this problem kinda just bounced off my head...

i was wondering if anyone could help me with this

[fluid tundra]

it would be good to consider how the (-1)^{k(k + 1) / 2} term behaves

[jade ice]

and what exactly does that mean

ohhh

meaning the parten?

that it generates?

[fluid tundra]

just try something

it is not like you have many options

[jade ice]

alright

ohhh i get it

it repeats as 1 raised to 2 - 2 raised 2 + 3 raised to 2......

and then we can just generate a pattern of pairs for 3 raised to 2 - 1 raised to 2 and same with 4 and 2

[mortal geode]

Telescopic series????

[jade ice]

which gives is the general formula of 4n(4n+1)

now i just need to verify it through answers

[mortal geode]

Close the channel now

U got the ans

@jade ice

Use .close

[jade ice]

yep i got the answer

.close

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[iron trail]

Find X

[odd edge]

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[frail lintel]

i thought it should havebeen sin^4 x

got it
.close

.close

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[ashen hatch]

Given $x^2+8xy-11y^2-16x+8y+16=0$, maximize the value of $x+y$.

[clever fjord]

laestia

[tepid pelican]

try doing some algebra

make some squares

[ashen hatch]

how do I do the thing

[tepid pelican]

hold on

[mystic nova]

this seems like some kind of (.....)^2 + (....)^2=0

[tepid pelican]

its actually quite weird

i think x + y is unbounded

yeah

[tacit haven]

is there a sol

[ashen hatch]

exactly 😭

[ashen hatch]

solution is apparently 4

[tacit haven]

x + y ends up depending on y

[ashen hatch]

idk how thats supposed to work tho

[tacit haven]

are you sure you typed it correctly

[ashen hatch]

yeah

[tepid pelican]

Choose arbitrarily large x, then I can simply choose large enough y which makes it 0

[ashen hatch]

x, y must be integers tho

[tepid pelican]

oh

now that's a different problem

[ashen hatch]

that's a req I forgot to mention

[tacit haven]

oh

😭

[ashen hatch]

or was it rational

[tacit haven]

if you want to be really bashy (and boring), treat the entire expression as a quadratic in x

[ashen hatch]

idk i don't know how to translate this

looks to be rational

Given $x^2+8xy-11y^2-16x+8y+16=0$, maximize the value of $x+y$ such that x and y are rational numbers.

[clever fjord]

laestia

[tepid pelican]

what chapter is this from?

[turbid parcel]

i figured it out

[tepid pelican]

its a degenerate conic (pair of lines)

[turbid parcel]

given the restriction that x and y must be rational, it's a disguised discrimnant question

where you wanna try to make a quadratic and find the value where you get a perfect square discrimnant, or 0

does that make sense?

[ashen hatch]

sub x+y=k and reduce it to a nice quadratic?

[celest osprey]

Looks like calculus

[turbid parcel]

yes

perfect

let y=k-x

[ashen hatch]

kk I'll try that

[turbid parcel]

and sub y

[ashen hatch]

yeah no I'm stuck

[turbid parcel]

whered you get

[ashen hatch]

-18x²+30kx-11k²-24x+8k+16=0

might've made an error with the calcs tho

[turbid parcel]

nope this is all correct

i'd reccomend rearranging for a positive x^2 coeff

my trackpad is acting up give me a second

I'm on phone now

[turbid parcel]

and then do this

[ashen hatch]

dang i really don't know what to do with this

am I supposed to do smth with 11k²

[turbid parcel]

what'd you get the discrimnant as

sorry my trackpad is being VERY finnicky

[ashen hatch]

I haven't even finished factoring it 😭

[turbid parcel]

ive been saying determinant im so stupid

i meant discrimnant

[ashen hatch]

discrim is b²-4ac right

[turbid parcel]

yes

the thing that goes under the root in the quadratic formula and thus determines whether roots are rational or not given integer coeffs

[ashen hatch]

OK I FIGURED IT OUT @turbid parcel THANK U

.close

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[hybrid vortex]

It's D right?

[mossy raven]

so i have found that my derivation of displacement is:
x = 8* sin(pi/5* t)-Vt

[mystic nova]

I use circle to solve this, tbh circle make this type of problem easy

[mystic nova]

This's not true

$x=8\cos(\frac{\pi t}{5}-\frac{\pi}{2})$

[clever fjord]

Alexis_Fx

[mossy raven]

how so?

[mystic nova]

if you use cos here, the beginning angle is -pi/2

[hybrid vortex]

I am not sure what type of formula should you use, but if the period is 10 seconds then that means it takes 10 seconds to go back and forth. Therefore, it takes 2.5 seconds to go half the distance to one side (or from the centre to 8m left/right and vice versa). So for 7.5seconds, you have gone once to the left, once back to the central point, and once to the right (total of three movements each by 2.5 sec). So you end up 8m to the right

[mossy raven]

sorry i meant to say sin,

[mossy raven]

im confused?

because since its initially moving to the left with V velocity, then the velocity would be:
v=8* pi/5* cos(pi/5*t)-V

because left is negative?

[mystic nova]

$v=8\frac{\pi}{5}\cos(\frac{\pi t}{5})$

[clever fjord]

Alexis_Fx

[hybrid vortex]

that's how i imagined it

[mystic nova]

you should imagine it with a spring

[mossy raven]

okay because 2.5*4 = 10

[hybrid vortex]

yeah

[mossy raven]

and the center of motion is in the middle right?

[hybrid vortex]

yeah

[mystic nova]

It's only true for a pendulum to be harmonic motion when the angle less than 10 degree

[mossy raven]

look dude this is maths

we assume anything

[hybrid vortex]

but why so

[mystic nova]

when the angle is less than 10 degree

We assume sin(x)=x

x is in radian

[mossy raven]

i think i know my mistake

[hybrid vortex]

I haven't learned any Physics, but how big of a change does it have when we assume it is still harmonic, even tho it's not

[mossy raven]

v should be 8/5* pi cos(8/5* t),
and v0 or V = 8/5* pi * cos(0)
= 8/5*pi

[mystic nova]

In physic?, it will make your whole solution wrong, we have different approach when the angle greater than 10 degree which is using energy

[mystic nova]

What do you think the maximum velocity is

[mossy raven]

i don't know? 8+8/5 pi?

[hybrid vortex]

but it is said that it moves in harmonic motion

it hasn't said anything about an angle

so i think it's right to assume that the angle is appropriate for it to be in shm

in this case below 10 degrees

[mystic nova]

Not much if the angle is less than 10 degree

It's ALMOST harmonic motion

[mossy raven]

What do you do with V?

[mystic nova]

if the angle's less than 10 degree

[hybrid vortex]

well in physics you round stuff up so

we say it is exactly, for the sake of simplicity

like pi = 3

and a cow is a sphere

[mystic nova]

@mossy raven let's analysis function v

$v=8\frac{\pi}{5}\cos(\frac{\pi t}{5})$

[clever fjord]

Alexis_Fx

[mystic nova]

using trig what do you think maximum value of this function is?

and when it occcupy

[mossy raven]

8* pi/5

[mystic nova]

yeah

And when?

[mossy raven]

At t=0?

Or when t=10k

[mystic nova]

well it's 2kpi

[mossy raven]

k being an integer

[mystic nova]

anyway

[mystic nova]

is this even right?

anyway

you know what V stated in the problem is

[mossy raven]

8/5* pi

[mystic nova]

Let see where it will be in 7.5 second

plug t=7.5

give you x=?

[hybrid vortex]

doesn't that give you v?

[mystic nova]

yeah

maybe plug t=7.5 to v function and see if it Vmax or 0

if it's Vmax or -Vmax then it's in the center

[hybrid vortex]

we get zero since cos(3pi/2) = 0

[mystic nova]

if it's 0 then it's at the border

[hybrid vortex]

shouldn't we use sin

[mossy raven]

It's negative 8

[mystic nova]

well then it's at center

wait

[hybrid vortex]

but how could it be

at the centre

[mystic nova]

you mean x or v

[hybrid vortex]

i think x

so you move in the other side (right)

[mossy raven]

X

[mystic nova]

if it's -8 it could be to the left, it's pretty much depend on the problem

-8 and 8 is the border

so v=0

[hybrid vortex]

it said it's initially moving to the left

so i think the minus would mean to the right

but again i think we use sin

oh no

v being zero means at the border

[mystic nova]

no it's to the left side, the angle move counter clockwise

[hybrid vortex]

the period is 10 seconds

no way it's to the left

[mystic nova]

hmm, hold on

[hybrid vortex]

just look at it with 2.5 sec movements

you move three times left, right, right

and you end up at the right border

[mystic nova]

yeah

[hybrid vortex]

can you check the answer

Boodeesa

[mossy raven]

The answer is D

[hybrid vortex]

great then

[mystic nova]

consider using circle for this type of problem

It will make it much easier

[mystic nova]

I just need to draw 3 line here and know where it end up be

[mossy raven]

I'm still confused on how to find the equation of displacement?

[mystic nova]

you mean x?

[mossy raven]

Yes

[mystic nova]

harmonic motion can be $x=A\cos(\omega t+\phi)$ or $x=A\sin(\omega t+\phi)$

[clever fjord]

Alexis_Fx

[mystic nova]

let say you choose the sin one

at the start x=0 that give us $\sin(\omega.0+\phi)=0$

[clever fjord]

Alexis_Fx

[mystic nova]

so $\sin(\phi)=0$

[clever fjord]

Alexis_Fx

[mossy raven]

Oh okay so there's no phase shift

Phi is zero

[mystic nova]

yes, you should only give $\phi$ value from $[-\pi,\pi]$

[clever fjord]

Alexis_Fx

[mystic nova]

$T=\frac{2\pi}{\omega}$

[clever fjord]

Alexis_Fx

[mossy raven]

So x=8sin(pi/5* t)

[mystic nova]

yup

hey, let do it with cos

$x=A\cos(\omega t+\phi)$

[clever fjord]

Alexis_Fx

[mystic nova]

what do you think phi is here

[mossy raven]

I'm just confused why it's not c)

Because at t=7.5 you get x=-8

And negative means it's to the "left" of the motion usually

[mystic nova]

I think it got shifted

[mossy raven]

Wait actually

I think left is the positive direction in this case

Because it says it's moving initially to the left

With v=V

Which implies v is positive which implies left is the positive direction

[mystic nova]

no

It always move to the left

itmove counter clockwise

I would be more surprise if it move to the right

[mossy raven]

On a circle yeah

But if you think of a pendulum it rocks from left to right

[mystic nova]

I'm thinking of a circle

[mystic nova]

like this one

[mossy raven]

Left from center point and right from a center point

Okay

I understand now

But I'm curious what the circle is

[mystic nova]

it represent harmonic motion in term of circular motion

When it moving from B' to B

The angle move from -pi to 0

[mossy raven]

Ah okay

So does it represent X and y as parametric sinusoidal equations?

[mystic nova]

I know what's going on here

Let say it slide from B' to B

so B = 8 and B'=-8

When we use sin, we shift this circle 90 degree

[mossy raven]

Okay cool

thanks

.solved

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[split cypress]

what am i supposed to do here?
Be the function f, f:R->R, f(x)= (x-1)[x], where [a] is the whole part of the number a. Then:
A. X=2 is a value of continuity for the function
B. X=1 is a value of discontinuity
C function is continuous on R
D x=2 is a value of discontinuity

[split cypress]

for both 1 and 2, the function is (0-1)[1 or 2]

they end up giving me -1 for x=1 and -2 for x=1

how do i go about doing the lateral limits to prove/disprove continuity where the function takes only one form

[manic sleet]

i'm not familiar with lateral limits, but you could just do a one sided limit from left and right and show they're not equal

[split cypress]

and thats where i would get the not whole part of the number to change, thus altering the final value depending on the side?

[split cypress]

i proved that both 1 and 2 aren't points of continuity, but the book answer is that only 2 is a point of discontinuity

wait i might have fucked up

oh wait the lateral limit influences the whole part

okay im dumb

.close

thx

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[thin rune]

would anyone be able to help me understand what this question is asking?

[novel hinge]

What do you understand about it so far?

[thin rune]

ig my understanding and attempt have more or less been, uncertainty(std deviation in this case ig) / actual value, I am just confused on what the actual value would be,

like is it?

0.3 / 2.32 or 0.3 / 41.76

I dont know why the give 18 pieces otherwise? but it might just be irrelevant?

[mental lotus]

do you know the actual formula to calclulate this quantity?

Like do you have it at hand for reference?

[thin rune]

theres an actual formula, thats just I was taught to do it in physics?

oh wait

[mental lotus]

Something along the lines of this

[thin rune]

I have never it expressed in that notation but that seems to be the general idea i was meant to be using

[mental lotus]

good

then you have 18 threads tested in this experiment

so that means 18 is the sample size

and you have the average as well as sd given

all you need to do is plug the values in appropriate places

[thin rune]

is this more binomial and stats formula, I cant find it listed as uncertainty

in which case, i dont think its the right approach?

[mental lotus]

this is a stats formula

thats why I asked you to confirm it

[thin rune]

this is more a physics question tho

like because its part of that class's practice problems

[mental lotus]

I dont think physics would have a different formula for this kind of a problem tho

look thru your notes and other stuff if you have any formula for this

[thin rune]

we usually do something like, value *sqrt(uncertainty(in this case sd maybe)/ actual value )

[mental lotus]

to find what?

[odd edge]

@thin rune Has your question been resolved?

[thin rune]

uncertainty?

[mental lotus]

value *sqrt(uncertainty(in this case sd maybe)/ actual value )
why do you have uncertainty in the formula to calculate the uncertainty?

[thin rune]

i think I explained it poorly but thats just how it works

[mental lotus]

ok idk then

[thin rune]

Ig I will just wait 15 before I can ping helpers and hopefully get told I am going in the wrong direction

[unkempt lichen]

are you perhaps looking to use the standard error of mean formula?

and is the uncertainty here the error of the mean?

(error in physics, not the erf - if you don't know what the erf is then ignore this)

[thin rune]

i think so, to me its just uncertainty in physics

[unkempt lichen]

then this seems to just be a plug and chug formula

do you have it?

[thin rune]

based on thats what the questions ask, i assume so

[thin rune]

if so, i dont have one, do you know where in a physics textbook or somewhere I would find one

the current course of this is flipped, so videos and then discussion and problems in class, I dont believe a formula was mentioned tho

[thin rune]

i definetly think i have heard this but not part of this lecture set up till this point so possibly not relevant, but can you tldr incase it is?

[unkempt lichen]

unfortunately, this is also a statistical formula

[thin rune]

erf is a statistical formula?

[unkempt lichen]

because this question is statistical by nature

erf is a different formula, connected to statistics

actually, more appropriate to call erf a function

its proper name is the Gaussian error function (or just error function)

you can ignore that function for now though

[thin rune]

what do I do if I dont have a formula tho?

[unkempt lichen]

are you not allowed to look for it?

because it literally just involves two terms

[thin rune]

i am, i just assumed it would be covered in flipped videos

but also everything uncertainty and physics comes up with stuff I already know?

[unkempt lichen]

i'm sorry, i don't get this question

[thin rune]

can I have the name of the formula so I can look it up myself because I cant find anything other calclated lab uncertainties?

[unkempt lichen]

it's called the standard error of the mean

[thin rune]

.close

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[pastel orbit]

so here's a proof I just wrote up.

I'm actually less interested in its correctness (though I'd still appreciate it if anybody is willing to review that!) and more in its structure

[pastel orbit]

I'm experimenting with trying a different proof structure, because I want to become more accustomed to saying "it suffices to prove xyz, here's why:" and then ending the proof once xyz is proven

to this end, is the way I've formatted this proof both readable and understandable?

I'm looking for feedback here so I can improve my writing

[pastel orbit]

and just in case, here're the definitions of a subbundle and a bundle homomorphism over M

but correctness is again, not my primary focus

[thin latch]

it was not about the destination , it was about the journey and friends we made along the way..

[odd edge]

@pastel orbit Has your question been resolved?

[tidal matrix]

i won't speak to the correctness because i havent taken diff geo properly yet ;-; i would like to share some tips on the structure as a fellow "wordy proof writer" (aka yapper):

\begin{itemize}
\item \textbf{line breaks / paragraph breaks are your friend:} with this much text, i think it's easy for someone skimming the text to be unsure what each sentence is doing. again, no hate on that, because i do it too B) but i think you could put paragraph breaks to delineate sections. \
I would put a paragraph break before "To this end..." and i would put a break before "In order to show this..." to make it clearer what you're doing. so the first paragraph is the short assumptions, the second paragraph explains why we are working towards xyz, then the last paragraph proves xyz.

\item \textbf{double dollar signs for important equations:} I think it helps the reader pick out the equations that are important if you center them. For example, I would center $$\pi_D = \pi_E \big|_D = \pi_E \circ \iota$$ because it is 1/3 of the smooth bundle homomorphism definition. I would also center something like $$\iota|D_p(D_p) = D_p = D \cap E_p \subseteq E_p$$. Personally, I'd also be inclined to center some of the longer equations at the bottom, but I think that's more personal taste and I don't see why you couldn't do that. Also note, centering equations automatically makes line breaks in the text which imo helps readability.

\item You have a bracket "(note that the image of...)" which goes on for about a line. This feels like less of a throwaway comment you're quickly sticking there, rather something of note that isn't directly relevant. I would thus recommend putting it in either a seperate sentence or maybe a remark at the end of the proof? It feels too long for me to be in a bracket is what I'm getting at.

\item The sentence "Now, note that it suffices..." is quite long - it takes up 5 lines! I would recommend splitting it.
\end{itemize}

[clever fjord]

haseeb

[tidal matrix]

hopefully those tips are helpful to you! let me write it out as a proof

oh good lord the latex

\textit{Proof.} The inclusion map $\iota$ is smooth by the definition of an embedded submanifold. Furthermore, by the definition of a subbundle, we have
$$\pi_D = \pi_E\big|D = \pi_E \circ \iota,$$
thus it remains to show that every $\iota$ is linear; i.e., that every $\iota\big|{D_p} = \pi^{-1}_D(p)$ is linear. \[2pt]
To this end, let $p \in M$ be arbitrary, and let $D_p = \pi^{-1}_D(p)$ be the associated fibre. We claim it suffices to prove that $$D_p = D \cap E_p.$$ This is because, as follows from the definition of a subbundle, $D_p$ is a linear subspae of $E_p.$ Additionally, $\iota\big|D_p$ is an inclusion map from a linear subspace to its ambient space, so these conditions impliy $\iota\big|D_p$ is linear.^\dagger \ ; \
In order to show this, let $d \in D_p.$ Then $p = \pi_D(d) = (\pi_E \circ \iota)(d) = \pi_E(d),$ so $$d \in \pi_E^{-1}(p) = E_p.$$
Conversely, suppose $d \in D \cap E_p.$ Then $\pi_D(d) = (\pi_E \circ \iota)(d) = \pi_E(d)$ which is simply $p,$ thus $$D \in \pi_D^{-1}(p) = D_p.$$ Since we have $D_p = D \cap E_P,$ then $\iota$ is linear. Since it is also smooth, it is a smooth bundle homomorphism over $M. \quad \blacksquare$ \ ; \ ; \

\textit{Remark.} ;^\dagger Note that the image of the $\iota\big|D_p$ will lie in $E_p,$ because $\iota\big|D_p(D_p) = D_p = D \cap E_p$ which is of course contained in $E_p$.

[pastel orbit]

hm, I do think I need to work on knowing when to break my paragraphs, but my entire proof here is only like 7 sentences - does that really require a line break?

the double dollar signs is probably fair

I just have a habit of doing intext lines

[tidal matrix]

up to you! i am probably a bit biased because i am reading ahlfors right now, and he has a horrible habit of starting theorems mid-paragraph. i think it does help clarify what each sentence is proving, exactly, so skimming it would be a bit easier?

[tidal matrix]

i have a habit of centering too much, honestly, but im sure you can find a healthy balance

[pastel orbit]

to the bracket part: I genuinely did not know how I'd write it out as anything but an additional comment

[clever fjord]

haseeb
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(You may edit your message to recompile.)

[pastel orbit]

the reason I have is because like, ${\iota|}_{D_p}$ is the restriction of $\iota: D \to E$ to the subset $D_p \subseteq D$

[clever fjord]

higher!

[pastel orbit]

but when you talk about a restriction map, the codomain remains the same, yeah?

so I need to justify why I'm allowed to restrict the codomain also

i.e. I need to show that the image lies in $E_p$

[clever fjord]

higher!

[tidal matrix]

ohh i see, so it is important to the proof and maybe not a remark

[pastel orbit]

but showing that $D_p = D \cap E_p$ suffices for this purpose, because then $D_p \subseteq E_p$

[clever fjord]

higher!

[tidal matrix]

i think the thing that got me was that you have an equation in brackets, and that seems like it's then too important to be contained in a bracket

[pastel orbit]

and the inclusion is the identity on D_p

oh hey!

congratulations on Helpful!!

[tidal matrix]

oh hey nice :D

ty

[pastel orbit]

the issue is that it's kinda not really relevant to the actual vector bundle theory

it's more or less purely a set theory thing, and something that I have no doubt would be ignored by most writers

written off as a detail the reader should figure out themselves

they'd just say the codomain can be E_p without saying why

it's the filling in of these details that's proven troublesome when it comes to writing for me

[tidal matrix]

how about "In particular, $\iota\big|D_p (D_p) = D_p$, so if $D_p = D_p = D \cap E_p \subseteq E,$ we can restrict $\iota\big|D_p$ to its codomain $E_p$ "

[clever fjord]

haseeb

[tidal matrix]

or if E_p is this ambient space, you could say "to its ambient space E_p"

[tidal matrix]

it could be worth bringing up in your proof, though! i think the rule is you are writing for an audience of your fellow peers, so if they needed clarification on this set theory topic, you should state it so they definitely have it

[pastel orbit]

hmm

I like this, actually

[pastel orbit]

in principle, my audience would be late UGs/early grad students, who probably wouldn't need smth like this spelled out so explicitly

but I have the tendency to write my proofs in a way that I'd be able to read without having to fill in details myself, and I'm kinda slow when it comes to these minor details, so..

[tidal matrix]

better to have it and be perceived as redundant, than to not have it and be perceived as wrong, in my opinion

it is ultimately up to you. i think any comment of that length, though, should ideally be its own sentence

[pastel orbit]

hm, at what point should somebody stop writing stuff like that then?

[tidal matrix]

but yeah you probably wouldn't need to state what a restriction map does for late UGs

[pastel orbit]

the textbook I'm reading has left out more major details before, and it's already known to be a very verbose book when it comes to its proofs

at some point I'll have to figure out what to leave out

[pastel orbit]

oh, lastly. yeah that "it suffices" sentence is really long haha

I have a tendency to write super long sentences with multiple clauses

maybe something to fix as well

[tidal matrix]

i once wrote a sentence that was a quarter of an A4 page long. i had loooots of editing to do that day

so i feel this pain

[pastel orbit]

[tidal matrix]

we can chalk it up to the editing stage hehe

[tidal matrix]

i think at the point where you become comfortable enough to know it, because you're writing for fellow students. like maybe once you get a few more chapters in, and you get "more used to" these ideas, those set theory connections will be second nature because you'll know the maps so well

but it's difficult because if you go back after being rusty, you might need those details again

i think unfortuntately it really depends on who you're writing the proof for and there's no good answer

[pastel orbit]

hm, I'm running into problems with the editing of that "it suffices" line..

Now, we claim that it suffices to prove that $D_p = D \cap E_p$. From this fact, it follows from the definition of a subbundle that $D_p$ is a linear subspace of $E_p$, and therefore that ${\iota|}{D_p}$ is linear, since it is being the inclusion map from a linear subspace to its ambient space (note that the image of ${\iota|}{D_p}$ will lie in $E_p$ because ${\iota|}_{D_p}(D_p) = D_p = D \cap E_p \subseteq E_p$).

[clever fjord]

higher!

[pastel orbit]

my problem remains with that note at the end

I can't say that the inclusion is a map from a subspace to an ambient space yet, because it's not yet stated that E_p really is the ambient space of D_p

that's what the note is for

but at the same time, I need to expain why the inclusion is linear

[pastel orbit]

and I can only do so in terms of saying the first thing here

so I can't tell what to say first

or if I need to change the sentence structure altogether

[tidal matrix]

ooh tricky! what about this:
"... from a linear subspace to E_p. However, (bracket thingy), so iota goes from a linear subspace to its ambient space."

or should it be to E, at first?

[pastel orbit]

wdym to E, at first?

[tidal matrix]

i.e. "from a linear subspace to E. However, ..."

because you can't quite yet say that iota restricts to E_p

if i understood you correctly, that is

[pastel orbit]

ah, so here's the thing

[sullen hollow]

Does anyone else have cameras on their bathrooms?

Or is it my family, only?

[pastel orbit]

${\iota|}_{D_p}: D_p \to E$ is set theoretically correct as a restriction map, right?

[tidal matrix]

wh

[sullen hollow]

[clever fjord]

higher!

[pastel orbit]

but the problem is that E is not a vector space

so it makes no sense to claim that this is a linear map

[tidal matrix]

welcome to mathcord! this is someone else's help channel, #discussion for... more "general" convos

[pastel orbit]

you need to restrict the codomain to E_p for it to even be possible that this map is linear

[sullen hollow]

thanks

okay

[pastel orbit]

It remains to show that $\iota$ is linear on each fibre of $\pi_D$, i.e. that ${\iota|}_{D_p}: D_p \to E_p$ is linear.

[clever fjord]

higher!

[pastel orbit]

that's why this line uses E_p

and not E

[tidal matrix]

oh yeah ofc

[pastel orbit]

but a priori, we don't even know that the image of D_p lies in E_p

[tidal matrix]

Now, we claim that it suffices to prove that $D_p = D \cap E_p$. From this fact, it follows from the definition of a subbundle that $D_p$ is a linear subspace of $E_p$. Then the image of ${\iota|}{D_p}$ will lie in $E_p$ because $${\iota|}{D_p}(D_p) = D_p = D \cap E_p \subseteq E_p$$. Therefore, ${\iota|}{D_p}: D_p \to E_p$ is linear, going from a linear subspace to its ambient subspace.

[clever fjord]

haseeb

[tidal matrix]

i.e. just move the comment up and treat it like an integral part of the proof

uh oh floating period strikes again

[pastel orbit]

hm, this is good

I do feel like the image line is clunky though, because it's not clear until the next line why we're suddenly discussing this

but maybe that's the best we can do

[tidal matrix]

right, and we wanted the "it suffices to show xyz. this is because"

[pastel orbit]

yeah

[tidal matrix]

Now, we claim that it suffices to prove that $D_p = D \cap E_p$. From this fact, it follows from the definition of a subbundle that $D_p$ is a linear subspace of $E_p$. Then if the image of ${\iota|}{D_p}$ lies in $E_p$, ${\iota|}{D_p}: D_p \to E_p$ is linear. In fact, $${\iota|}{D_p}(D_p) = D_p = D \cap E_p \subseteq E_p,$$
so $\iota |_{D_p}$ is a linear map from a linear subspace to its ambient subspace.

[clever fjord]

haseeb

[tidal matrix]

we can also put the equation inline

[pastel orbit]

Now, we claim that it suffices to prove that $D_p = D \cap E_p$. From this fact, it follows from the definition of a subbundle that $D_p$ is a linear subspace of $E_p$. However, because the image of ${\iota|}{D_p}$ will then also lie in $E_p$ as ${\iota|}{D_p}(D_p) = D_p = D \cap E_p \subseteq E_p$, we have that $E_p$ is the ambient vector space of $D_p$. Thus, ${\iota|}_{D_p}$ would be the inclusion map from a linear subspace to its ambient space, and therefore linear.

[clever fjord]

higher!

[pastel orbit]

hm, this is what I just edited it to

Now, we claim that it suffices to prove that $D_p = D \cap E_p$. From this fact, it follows from the definition of a subbundle that $D_p$ is a linear subspace of $E_p$. However, because the image of ${\iota|}{D_p}$ would then also lie in $E_p$ (since ${\iota|}{D_p}(D_p) = D_p = D \cap E_p \subseteq E_p$), we'd have that $E_p$ is the ambient vector space of $D_p$. At last, it follows that ${\iota|}_{D_p}$ would be the inclusion map from a linear subspace to its ambient space and therefore linear. With this in mind, we focus our attention on proving this set equality.

[clever fjord]

higher!

[pastel orbit]

another edit?

[tidal matrix]

ooh i do like that

[pastel orbit]

I turned the equality chain back into a quick remark

this proof structure is so scuffed

[tidal matrix]

it doesn't explicitly follow the "xyz because" structure, but i think it works and is very smooth

probably because 2 people are discussing it at once

i think altogether that works, and the wordiness honestly helps understand what's going on.

more words in proofs!!

[pastel orbit]

that's fine

I like the clarity of this honestly

[pastel orbit]

I will continue to be a yapper then

this one remark literally makes up half the proof now though LOL

okay, edited once again

I realized that I didn't need the "E_p is the ambient vector space" part because that follows directly from the "D_p is a linear subspace of E_p" @tidal matrix

Now, we claim that it suffices to prove that $D_p = D \cap E_p$. From this fact, it follows from the definition of a subbundle that $D_p$ is a linear subspace of $E_p$. However, because the image of ${\iota|}{D_p}$ would then also lie in $E_p$ (since ${\iota|}{D_p}(D_p) = D_p = D \cap E_p \subseteq E_p$), we'd at last conclude that ${\iota|}_{D_p}$ is the inclusion map from a linear subspace to its ambient space and therefore linear.

[clever fjord]

higher!

[pastel orbit]

this is the line now

eh, maybe I don't like the at last part

I'll delete it

just those two words

Now, we claim that it suffices to prove that $D_p = D \cap E_p$. From this fact, it follows from the definition of a subbundle that $D_p$ is a linear subspace of $E_p$. However, because the image of ${\iota|}{D_p}$ would then also lie in $E_p$ (since ${\iota|}{D_p}(D_p) = D_p = D \cap E_p \subseteq E_p$), we conclude that ${\iota|}_{D_p}$ is the inclusion map from a linear subspace to its ambient space and therefore linear.

[clever fjord]

higher!

[tidal matrix]

i think this is the one! it's super clear and it flows nicely between all the different things

and the sentence is good either way, although i do like it without the last two words

it does get a touch verbose there, but very well done

[pastel orbit]

thank you for all your feedback, this was super helpful

doing this alone would've been really painful

dang it

how many edits

LOL

[tidal matrix]

of course! i hope i was helpful, keep yapping in your proofs it makes the world go round

[pastel orbit]

okie dokie, I'll close this now

.solved

[odd edge]

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[pastel orbit]

congrats on Helpful!!

[odd edge]

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[mystic saffron]

HI

[mystic saffron]

does e or ln or cos or sin etc has complex roots ?

[quasi sparrow]

if their arguments are complex, then yes

[mystic saffron]

ive e to power complex function

[quasi sparrow]

oh then no not the exponetial

[mystic saffron]

how about tringomatric fucntions ?

[quasi sparrow]

,w solve e^z = 0 for complex z

[clever fjord]

Results provided by WolframAlpha

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[brittle beacon]

(worth noting the latter three have real, hence complex, roots)

[wooden python]

trig(z)=0 has no solutions beyond the real ones

[quasi sparrow]

oh yes if sin(z) = cos(z) = 0 then z must have 0 imaginary part

[wooden python]

sin(z)=cos(z)=0

[mystic saffron]

is this web solution 100% correct ?

[brittle beacon]

There are no real or complex solutions to e^z = 0, yes

[odd edge]

@mystic saffron Has your question been resolved?

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[wanton bison]

. @austere dragon

[austere dragon]

ye

[austere dragon]

oh yay

thx

[pliant nexus]

uh what is $\sqrt{9}$

[clever fjord]

Arnavutköy

[austere dragon]

what D =

[pliant nexus]

the way to solve this is slowly to remove things from the left hand side until you are left with D

[austere dragon]

ok

tbh i dont know how to do log lol

my friend trying to quiz me

[stark heart]

Take it step by step
First what is √9

[austere dragon]

3

[stark heart]

Ok so how to get it to the rhs

[austere dragon]

yes'

nvm guys i used chatgpt

its 8=D

wait

why does that look like a pp

my friend trollin me

[stark heart]

It is nice that you got the answer
But it would be better to be able to solve it yourself

[austere dragon]

ok

[stark heart]

What would you do if you got a similar question on a quiz or in an exam

[austere dragon]

calculator

or google translate

[stark heart]

The "Show your steps"

[austere dragon]

um idk

[stark heart]

So how to get the +3 to the other side?

[austere dragon]

wym

oh nvm

[stark heart]

We currently have
$$\log_2(2^3\cdot \frac{D}8)+3=6$$

[clever fjord]

Sherif Player

[stark heart]

How can we remove the 3 from the LHS or how to move it to the RHS

[austere dragon]

subract

[stark heart]

If you're satisfied with chatgpt answer then there is no need for me to explain

[austere dragon]

ok

[stark heart]

!done

[odd edge]

If you are done with this channel, please mark your problem as solved by typing .close

[austere dragon]

.close

[odd edge]

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[austere dragon]

.skibidi

[odd edge]

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[vagrant oxide]

Aaaaaaaa

[crystal charm]

hi, what's your question you need help with?

Alright I guess you just wanted to jumpscare us with the "Aaaaaaaa"

[odd edge]

@vagrant oxide Has your question been resolved?

[odd edge]

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[rare roost]

I keep getting this wrong

I’m not sure what I’m doing wrong

This is the question

[rough birch]

Did you flip the - 45 to a +45?

[rare roost]

im not to sure what im doing tbh

[rough birch]

Okay, we know we want

sin(x-45) = sqrt2 /2

let’s let u = x - 45 for now and just find all the solutions

when does sin(u) = sqrt2 /2 in general

[rare roost]

45, 135,

[rough birch]

yup and just those +- multiples of 360 right

[rare roost]

yes

[outer cradle]

Math

[rough birch]

now we want x - 45 = (45 or 135) + k * 360, where k is any integer

[rare roost]

mhm

[rough birch]

if we add 45 to both sides we get

x = (90 or 180) + k * 360

[rare roost]

yes

[rough birch]

and now we just need to find the right k value for each solution to be within our interval

here it’s pretty simple

[rare roost]

domain 45 - 405??

45<x+45<405

???

[rough birch]

I think you’re over complicating it, we already did the hard bit

We solved for all solutions for x in general

Now we just need to find the ones between 0 and 360

[rough birch]

If these are our solutions, which ones are within the domain we want (0 <= x <= 360)

[rare roost]

i’m not to sure

[rough birch]

let’s start with the 90 solution

x = 90 + k * 360 for any integer k

Which value(s) of k put x between 0 and 360

[rare roost]

uhm

[rough birch]

what does x equal if k is 0

[rare roost]

what does *360 mean?

[rough birch]

times 360

if you add multiples of 360 degrees to an angle, the trig expression won’t change right

[rare roost]

90

x = 90 when k =0

[rough birch]

yes exactly!

[rare roost]

so one solution is 90??

[rough birch]

yup

And 90 is within our domain too so it’s perfect

are there any other solutions that lie in our domain of 0 to 360?

[rare roost]

180

[rough birch]

yes, that’s from the 180 solution

And we can’t add or subtract 360 from either of these because then we would be either below 0 or above 360

so our only solutions are 90 and 180

[rare roost]

ohhh ok i understand

now

ok that actually makes sense now

thank you

.close

[odd edge]

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[pseudo island]

So I decided to take a de college algebra but the thing is I'm shit at algebra. Def cuz I wouldn't pay attention and would fall asleep in algebra 1 and that messed me up in algebra 2. But I pulled through and end 2 with a 91. (Bad but it could've been worse so) havnt done any math in like a yr and I forget easily. Tried khan academy but it's boring the videos make me sleepy and the question seem repetitive. Don't like it, need better studying stuff cuz I need to relearn this stuff I'm trying get good grades to boost my GPA for the algebra fumble

[crystal charm]

What's your question though?

Also is english not your native language, I had a bit of trouble reading that

[crystal charm]

oh nvm you're asking for better studying stuff

[pseudo island]

Yh, English is my 2nd language

[crystal charm]

I guess you could try textbooks, if watching videos is not your thing?

[pseudo island]

I could but they ain't cheap. Also not the best visual learner. I'll give it a try tho, u know any online textbooks? (download able preferred)

[crystal charm]

I don't really know textbooks for algebra if i'm going to be honest with you, you can probably google or ask someone else

[simple epoch]

Search algebra textbooks openstax. Openstax hosts free textbooks written by professors so they’re probably your best bet

[crystal charm]

If it's for calculus, I've heard stewart's is pretty good

I've never personally used it though

[pseudo island]

That too taking pre calculus next semester

[simple epoch]

This is the textbook my university uses for algebra, https://academics.uccs.edu/kcliff/collegealgebra/
They also have online lecture videos, iirc nonstudents can access those, lemme check
Edit: not for college algebra sadly.

[odd edge]

@pseudo island Has your question been resolved?

[odd edge]

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[pseudo island]

Thanks for all the help guy sorry for responding late.

[pseudo island]

@crystal charm Thanks

Both of yall are life savers

[crystal charm]

np

[pseudo island]

.close

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[shrewd trellis]

guys im so lost

[shrewd trellis]

what is this even saying

[unkempt lichen]

this seems like the definition of a definite integral

[shrewd trellis]

hmm

what about the annotations

my lecturerr made

[meager juniper]

A sketch of the proof of the Fundamental Theorem of Calculus

[unkempt lichen]

!occupied

[odd edge]

Someone else is already using this help channel. If you need help with a question, please open your own help channel/thread (see #❓how-to-get-help for instructions).

[meager juniper]

Was that in response to me?

Or did someone post something and then deleted it?

[unkempt lichen]

no, there was another person who posted their question here

he's in #help-12 now

sorry

[shrewd trellis]

wait whats happening

[unkempt lichen]

nah someone just came in and left

it's fine

[shrewd trellis]

oh

[shrewd trellis]

whats a proof?

[unkempt lichen]

a proof is how you show a statement is true

or false, depending on context

but in this case, we want to show why the statement is true

[shrewd trellis]

ohhh ok

I don't really understand

the dx

in this context

[unkempt lichen]

dx (or d-whatever) tells you the variable that is being integrated here

if it's da, then the integral is with respect to a

if dn, then the integral is with respect to n

[shrewd trellis]

could u show me an example?

[unkempt lichen]

of what

[shrewd trellis]

of a variable being integrated

[unkempt lichen]

here

[shrewd trellis]

u didnt put

upper and lower bound on the

s

[unkempt lichen]

indefinite integral

[shrewd trellis]

symbol

oh

[unkempt lichen]

doesn't matter, the concept of variable of integration is the same

[shrewd trellis]

what is a indefine integral

[unkempt lichen]

when you don't have upper/lower bounds

[shrewd trellis]

ohh

[unkempt lichen]

i mean if it helps you we can put some random bounds

but it doesn't change my point here

[shrewd trellis]

so it applies to the whole function

[unkempt lichen]

wdym

[shrewd trellis]

like what does an indefinte integral do

[unkempt lichen]

if you think about integrals being the area under a curve

definite integrals give you a definite answer about the area under a curve

indefinite integrals simply give you a way to find the area under a curve
but because no bounds are set, no area can be defined

but i'm curious. have you not faced these terms before? you're already facing definite integrals, and it's a bit interesting if you have not met even a single indefinite integral yet by this point

[shrewd trellis]

so u could be finding area infinitily

[unkempt lichen]

not infinitely

that is a whole other story

indefinite integrals result in functions

definite integrals usually result in numbers

[shrewd trellis]

example?

[unkempt lichen]

can you integrate this?

[shrewd trellis]

x^2/2

[unkempt lichen]

you're missing something

[shrewd trellis]

+c

[unkempt lichen]

there we go

but from this can we tell the area under this curve?

[shrewd trellis]

u cnt put

the numbers

[unkempt lichen]

of course we can't, right? we didn't specify the bounds

[shrewd trellis]

and sub it

on the integrand

since there isnt

any bound

[unkempt lichen]

yes, but we know that if we do have bounds, we can find the area under the curve between those bounds using this formula

or rather, this function

[shrewd trellis]

which formula/

?

u mean function

[unkempt lichen]

this

[shrewd trellis]

oh

ok

fair

[unkempt lichen]

suppose we now have bounds though

that means we now have a way to find the area under this curve (or line), because we have a solid area to work with

that's the difference between indefinite and definite integrals

[shrewd trellis]

yea so with

undefinite integrals

you can't solve an area?

[unkempt lichen]

I mean

let's say we have this curve

by the nature of indefinite integrals, we don't have bounds

so which area are we talking about?

this one?

this one?

or perhaps even this one?

that's why they are called indefinite

definite means that there is a concrete answer

[shrewd trellis]

so it goes forever

[unkempt lichen]

it doesn't go forever in that sense

[shrewd trellis]

or indefine is just

too vague?

for there to be one answer

[unkempt lichen]

i'd rather you think that indefinite is too vague, yes

because, again, we don't have bounds on where to start measuring the area from and to

it's like asking someone to measure the length of a section of the road

without giving them some landmarks to start and end at

where is the guy supposed to measure? which section?

but they know that if they want to measure the length of this road, they would need some measuring tape

indefinite integrals are this measuring tape. definite integrals give you those start and end points

but later in your studies you might meet integrals being used to calculate volumes

[shrewd trellis]

wdym indefine integrals are this measuring tape

[unkempt lichen]

that's a story for another time though

[shrewd trellis]

yeah exactly

[unkempt lichen]

as mentioned, indefinite integrals give you the function you can use to calculate the area under a curve

much like a measuring tape gives you the ability to measure the length of a section of the road

[shrewd trellis]

oh yes right

[woven cedar]

indefinite integral of a function makes it so you can calculate any area within two x values under the function by just plugging them in as upper and lower limits

also called antiderivative and used as the opposite of deriving in some questions

Hana did a fantastic job explaining it

[unkempt lichen]

took me like four tries though hahah, wouldn't consider this fantastic

[shrewd trellis]

wdym?

is this what hana explained

or new info?

[woven cedar]

lets take the function y=x²

whats the indefinite integral of it

[shrewd trellis]

you did an amazing job

[woven cedar]

it pretty much is yeah

I just wanted to add a few things

[unkempt lichen]

in short, indefinite integrals are the tools to measure the area under a curve. definite integrals tell you where you should measure that area

[tidal matrix]

$$\underbrace{\int^b_a f(x) ; \dd x}{\text{specific area \ under curve, \ from a to b}} = \underbrace{F}{\text{measuring tape}}(\underbrace{b}{\text{applied to }b}) - \underbrace{F}{\text{measuring tape}}(\underbrace{a}_{\text{applied to }a})$$

oh good lord

that's even WORSE

ok bad formatting but maybe you can see the idea that hana was saying, in terms of the fundamental theorem

[woven cedar]

@shrewd trellis you should watch visualisation vids on yt

[clever fjord]

haseeb

[tidal matrix]

(the cutoff says "applied to a")

[woven cedar]

hell yeah

amazing

[unkempt lichen]

here's a simple visualization

[woven cedar]

this applies to any b and a value as long as the function is defined in that interval

even if you have some weird partial function

[unkempt lichen]

not only must the function be defined

it must be continuous

[woven cedar]

yeah

[unkempt lichen]

otherwise this theorem will not work

[amber schooner]

thomaes function?

[amber schooner]

no

are step functions not integrable?

[woven cedar]

popcorn

[unkempt lichen]

in relation to the fundamental theorem of calculus

i mean sure step functions are integrable, but i don't think that's the question here, no?

[amber schooner]

you said f needs to be continuous

or are you saying the anti derivative is

[unkempt lichen]

f must be continuous in the interval [a, b] for the fundamental theorem of calculus to apply, doesn't it?

[amber schooner]

which FTC?

[unkempt lichen]

the one shown in the image

[woven cedar]

F(b)-F(a)

[amber schooner]

oh

[shrewd trellis]

what did I miss?

[woven cedar]

dude u should seriously watch some visualisations

so u can get the concept

[shrewd trellis]

fundamental thereoum of calc 1?

[woven cedar]

the concept of integrals

[unkempt lichen]

tbh the moment you were confused about indefinite integrals, you should probably revise that

https://www.youtube.com/watch?v=rfG8ce4nNh0&pp=ygUVM2JsdWUxYnJvd24gaW50ZWdyYWxz
here's a good vid to start

[woven cedar]

indefinite integral is just the short way, mathematicians just find shorter ways to do stuff

same way how you dont take limits everytime you calculate derivatives

its just essentially 2 tangent points very close to eachother

you take the limit

[odd edge]

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[naive sierra]

what did i do wrong

the answer key has 2 solutions

x=2 and x=-6

[wooden python]

$4^{x^2} = 4^{(x^2)} \neq (4^x)^2$

[clever fjord]

Ann

[wooden python]

$(4^x)^2 = 4^{2x}$ rather

[clever fjord]

Ann

[wooden python]

so you misread the equation and ended up with a completely wrong working

[naive sierra]

woops

hmm

tysm

i understand

.close

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[manic grail]

Hello im struggling with limit of f/g proof

[manic grail]

I did this but cant see how to control 1/|g(x)| idk if i should have taken a different approach

[wooden python]

you should use the fact that l ≠ 0 somehow

[manic grail]

Well i need an opposite inequality I think

|g(x)| > of something

[wooden python]

there exists a nbhd of x_0 in which |g(x)| ≥ |l|/2

[manic grail]

Can i say |g(x)| > 0?

[manic grail]

Why?

[wooden python]

you can but it won't help.

[wooden python]

epsilon-delta defn with epsilon = |l|/2.

[manic grail]

Wdym nbhd?

[wooden python]

nbhd = neighborhood

[manic grail]

Wait but

Isnt it g(x) > l/2

How we get the absolute value

[wooden python]

i mean... l can be negative, no?

im covering my bases here

[manic grail]

Yes exactly

So how do we know |g(x)|> |l|/2

[wooden python]

gimme a sec to write it out

[manic grail]

Alr

[wooden python]

this is the formalism behind it

the informal idea is that g(x) is separated from 0

in some nbhd of x_0

[manic grail]

Oh wait

ε= |l|/2

Thought you said ε = l/2

Ok yes i see

So like this ?

@wooden python

[wooden python]

honestly dont feel like reading all that rn sorry

[manic grail]

Nws

[manic grail]

If someone could review this i think its correct but not sure if i made any mistake

[odd edge]

@manic grail Has your question been resolved?

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[mystic saffron]

I have so far been able to find that (log(a))^3/2 = (3 +sqrt(5))/2 and (log(b))^3/2 = (3-sqrt(5))/2

[mystic saffron]

I am using a for alpha and b for beta

Please tag me

[narrow crypt]

[ \log_\alpha \beta \cdot \log_\beta \alpha = 1]

[clever fjord]

k

[narrow crypt]

would cubing this work

[wispy river]

it works

[narrow crypt]

[ 3(\log_\alpha \beta)^{4/3}(\log_\beta \alpha)^{2/3} = 3\log_\alpha\beta^{2/3} \cdot 1]

[clever fjord]

k

[narrow crypt]

wonderful

ye it works

cool

[wispy river]

(a+b)^3 = a^3 + b^3 + 3ab(a+b) this form helps

[narrow crypt]

@mystic saffron

[mystic saffron]

yes that works

[mystic saffron]

is there a way to move forward using this piece of information

[wispy river]

isnt it to the power 2/3 and not 3/2

You can cube and add them

[wooden python]

was nobody going to suggest putting x := log_alpha(beta)^(1/3) or sth

[wispy river]

using log properties is easier here

[wispy river]

I think op has found the roots already using this method

[meager palm]

I have to do 5 chapters in a few hours for my maths exam tmr what shuld i do? 😭

[mystic saffron]

i did x := log_alpha(beta)^(2/3) to find the roots

[wispy river]

thats why you got 3+root5/2 and 3-root5/2 right

[mystic saffron]

yes

[wispy river]

you can cube them to get the answer as well

[mystic saffron]

wait let me try

[narrow crypt]

lock the tf in and !occupied

!occupied

[odd edge]

Someone else is already using this help channel. If you need help with a question, please open your own help channel/thread (see #❓how-to-get-help for instructions).

[mystic saffron]

I GOT IT!

4 again

thank you so much!

i appreciate it!!!

.close

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[tiny pendant]

first i got this

the only valid inequality which still allows the above expression to hold true are x=y>z and x=y=z

right>

so whats the final answer

[wooden python]

lemme double-check that...

[tiny pendant]

im not making any progress after this

216000=2^6 x 3^3 x 5^3'

[wooden python]

so it's either x=y=z or x<y=z or y<x=z or z<x=y

i.e. two are equal and the third is lower, or all 3 are equal

000 111 222 333 444 555 666
011 101 110
022 122 202 212 220 221 
033 133 233 303 313 323 330 331 332
044 ...

[tiny pendant]

yea

so how will we get to the final answer

[wooden python]

well, you break a, b and c into their prime factorizations

$a = 2^{x_1} 3^{x_2} 5^{x_3}, b = 2^{y_1} 3^{y_2} 5^{y_3}, c =2^{z_1} 3^{z_2} 5^{z_3}$

[clever fjord]

Ann

[wooden python]

(x_1, y_1, z_1) must be a trio satisfying the inequality condition you worked out and with all numbers between 0 and 6 inclusive

(x_2, y_2, z_2) and (x_3, y_3, z_3) must be the same but with the numbers between 0 and 3

[odd edge]

@tiny pendant Has your question been resolved?

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[slow cradle]

for this

[slow cradle]

i dont get

so at the start

[minor bronze]

you mean the start?

[glass vault]

You expanded the denominator wrong

$(x^a)^b=x^{ab}$

[clever fjord]

ImOakley

[slow cradle]

why

can i not just

[slow cradle]

thats not me

so basically

can i not just simplify the 3^x at the start

[minor bronze]

im pretty sure $ 2^{9^x} $ is not equal to $2^{(9^x)}$

[clever fjord]

hecker

[slow cradle]

bro idrk

i got it wrong mysel

i tried to make the the numerator and denominator

have 3^x

[glass vault]

Oh wait its $3(3^x)=3^{x+1}$

[clever fjord]

ImOakley

[glass vault]

My bad

[slow cradle]

so i made the 9^x= (3^2)^x=(3^x)^2

is that not valid

[glass vault]

Yeah thats fine

[slow cradle]

ok

so why do we have to

make the numerator and denominator

have 2 as a base

[glass vault]

Because then you can simplify it

[slow cradle]

can i not

just leave it

[glass vault]

$\frac{x^a}{x^b}=x^{a-b}$

[clever fjord]

ImOakley

[slow cradle]

can we even do that

in this case

[glass vault]

Yes

[slow cradle]

we have like a power

to a power

[glass vault]

Doesnt matter

[slow cradle]

bro i genuinelyt

dont even know

if im good at maths

im getting like

every question wrong

its insane