the other one is q
#help-36
pliant shore
thin night
Did my teacher skip on those cuz I’ve only seen him write things from textbook to the board only-
pliant shore
Did my teacher skip on those cuz I’ve only seen him write things from textbook t...
maybe
some teachers don't explain properly
thin night
He speaks quietly and just writes the problems from textbook on the board and solve it himself and make us read the rules written on the textbook so I get very sleepy in his classes-
pliant shore
sigh, do Japanese teachers still teach like that....
thin night
Had a better strict grandma teacher but she was changed to middle school teacher from my highschool-
This male teacher also always busy and gives us lot of study periods while he handle his jobs outside school-
And when you ask him question he tells us to figure it out ourselves and call it easy
When I told him it’s hard and I can’t understand anything he straight up gave me weird look and told me it’s easy
And now his wondering why I’m failing his class
thin night
Now test is tomorrow and now I’m here struggling to understand anything from textbooks
pliant shore
right, back to the question I guess
thin night
Kk
pliant shore
right, back to the question I guess
but then the original y-value (sin is y) is negative, but the new y-value is positive
so you have $\sin(7pi/6) = - \sin(pi/6)$
soft zealotBOT
south
thin night
I’m having hard time understanding
pliant shore
right, back to the question I guess
do you agree that a straight line connects these two points?
so the angle on a straight line is 180 degrees = pi radians
thin night
do you agree that a straight line connects these two points?
Yeah I guess
thin night
so the angle on a straight line is 180 degrees = pi radians
Ohh
final saddleBOT
@thin night Has your question been resolved?
final saddleBOT
final saddleBOT
Available help channel!
fossil mauve
i need help with understanding hinge theorem, and also with this problem😢
final saddleBOT
i need help with understanding hinge theorem, and also with this problem😢
@fossil mauve Has your question been resolved?
fossil mauve
.close
alpine compass
$$\left( \sum_{n=0}^{\infty} \frac{1}{n!} \right)^{4 \sum_{k=0}^{\infty} \frac{(-1)^k}{2k+1} , \sqrt{-1}} + \sum_{n=0}^{\infty} \frac{n!}{(n+1)!} = 0
$$
soft zealotBOT
oguz
final saddleBOT
$$\left( \sum_{n=0}^{\infty} \frac{1}{n!} \right)^{4 \sum_{k=0}^{\infty} \frac{(...
alpine compass
could someone explain this😭
green pebble
the sum in the power is pi/4
so the whole power simplifies to i * pi
the sum in the base is a definition for e
green pebble
yes
alpine compass
right thanks
woven ledge
harmonic series
elder badge
somber hare
May I ask a question?
woven ledge
May I ask a question?
final saddleBOT
@alpine compass Has your question been resolved?
shrewd wren
I need help with the task below:
We can write $e^x$ as an infinite series called a Taylor polynomial: $$e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}= 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + ... $$
By including a finite number of terms in the series, we obtain an approximation of $e^x$.
Find the integral $I$ by approximating $e^x$ with a Taylor polynomial with four terms.
final saddleBOT
I need help with the task below:
We can write $e^x$ as an infinite series calle...
soft zealotBOT
dream
vital crag
Where's the definition of I
shrewd wren
$I = \int_{0}^{1}e^x dx$
soft zealotBOT
dream
worldly mesa
Then understand how large the error is
vagrant smelt
question, how can I the the x value of local min/max and the y value of global max/min
final saddleBOT
question, how can I the the x value of local min/max and the y value of global m...
@vagrant smelt Has your question been resolved?
Available help channel!
final saddleBOT
Well need a lil help in this limit
tranquil pine
ill share what i have done
tired walrus
.
for future reference: starting a message with . PREVENTS channel opening
also did you mean to upload a pic of your question?
tranquil pine
also did you mean to upload a pic of your question?
yes well takes a bit time since i do not have the app on my phone im sorry
tranquil pine
for future reference: starting a message with `.` PREVENTS channel opening
thanks for the info
tired walrus
ok let's wait for the question and your work then
tranquil pine
1st try
then some sin cos weirdo just to separate the 1st n second try
the rest of the 2nd try
ح ع ت means indeterminate
tired walrus
tranquil pine
yes
tired walrus
half of your x's look like n's. it's a bit disorienting
tranquil pine
half of your x's look like n's. it's a bit disorienting
well ouch sorry its not he first time i get this remark
tired walrus
so... i see you're trying to do it with l'hôpital or something. is that a requirement?
or is that just what you chose
tranquil pine
so... i see you're trying to do it with l'hôpital or something. is that a _requi...
idk whats l'hopital actually
tranquil pine
ouh well yeah if it is called like that i thought it has smth to do with solving a limit using derivation
tired walrus
im asking if this is:
A) the method your teacher requires you to do
B) simply the method you chose yourself for this question
C) the only method you know anyway
tranquil pine
im asking if this is:
A) the method your teacher _requires_ you to do
B) simply ...
B) and C)
since i tried all what i knew could help
or thought
tired walrus
ok right
can i suggest an alternative?
it's not going to involve any derivatives at all
tranquil pine
should i revise perhaps i have forgotten smth
or its solution acquires a certain other method
tranquil pine
it's not going to involve any derivatives at all
i would love to yeah
tired walrus
ok right
i think this will be easiest if i write it on paper cause i kinda don't feel like doing it in LaTeX
so gimme a few mins
tranquil pine
i think this will be easiest if i write it on paper cause i kinda don't feel lik...
ayyy i understand
take ur time :DDDDD
tired walrus
this is only part of the solution bc im not giving out the full one
but this is how i thought to do it
you tried to start off with splitting the fraction too but your split went too far
if you do it how i showed, you can still break off a part that is safe to put x=2 into
and then take care of the other part separately
tranquil pine
this is only part of the solution bc im not giving out the full one
I'm processing/understanding each step and try to re do it alone to make sure i got it fully
I'm really thankful
tired walrus
you can and should ask me about specific steps if you have doubts
tranquil pine
if you do it how i showed, you can still break off a part that is safe to put x=...
a safe part is the one that gives a real number right?
tired walrus
"safe" in this case means doesn't cause a division by 0
tranquil pine
ouh god
the only thing i would ask
how would u know which part from the separated root/sqrt to leave under that frac that is between ()
or ull try to imagine how it would be like later
tired walrus
how would u know which part from the separated root/sqrt to leave under that fra...
i pulled the 1/sqrt(x+2) to the side bc that's a "safe" factor -- putting x=2 into it will give you 1/2
but you can't substitute x into only part of the function you're taking the limit of
so that just stays there as-is until the rest is ready to have x substituted in
(sqrt(x)+sqrt(2)) is also a safe factor btw. could be pulled aside in a similar way but what you did still has the correct key idea
so yeah you have the correct answer of -1/2 and you captured the process correctly
dense palm
@tired walrus how did you get so good at math
tranquil pine
<@183668144404037632> how did you get so good at math
frrrrr like omg
final saddleBOT
@tranquil pine Has your question been resolved?
tired walrus
<@183668144404037632> how did you get so good at math
i mean... i have a master's degree in it AND i work as a math teacher.
AND i studied at a relatively prestigious uni in my country of birth
AND i was lucky to have parents who are on good terms with math
so like idk. it's a whole bunch of stars all aligned in my case
@tranquil pine in case you're also interested
shut minnow
Can you solve the reimman conjecture tho /jk
final saddleBOT
final saddleBOT
Available help channel!
elfin fog
Can I have help with this question please?
final saddleBOT
Can I have help with this question please?
What step are you on?
1. I don't know where to begin.
2. I have begun but got stuck midway.
3. I got an answer but I was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above
3
tribal ocean
What do you know about $x$ and the frequency density shown by the upper red line
soft zealotBOT
flynger
final saddleBOT
@elfin fog Has your question been resolved?
heady reef
how would i find the horizontal phase shift for this graph?
final saddleBOT
how would i find the horizontal phase shift for this graph?
@heady reef Has your question been resolved?
What step are you on?
1. I don't know where to begin.
2. I have begun but got stuck midway.
3. I got an answer but I was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above
2
i've got the amplitude, period, and the vertical shift
just need the horizontal phase shift
hoary tiger
recall that sin(0)=0
and that the amplitude of sin(x) is y=0
where does there exist a similar point on your function (where f(x)=amplitude)
do keep in mind what sin(x) normally looks like
heady reef
The amplitude for that graph is -1.72
When x = 2.35 y = -1.72
I could use that point
hoary tiger
yep :)
final saddleBOT
final saddleBOT
Available help channel!
ember river
I'm having trouble identifying the sample versus population and it's stressing me out
final saddleBOT
I'm having trouble identifying the sample versus population and it's stressing m...
ember river
Technically the pesticides are being tested, not the crops?
cloud pawn
I'm having trouble identifying the sample versus population and it's stressing m...
population = everyone/every thing
sample = portion of the population
its been a year since i took AP stats
ember river
population = everyone/every thing
sample = portion of the population
its been a...
So
that makes it sound like ALL the crops are population
and just the 10 of each of his crops are the sample
I'm just unsure because it's the pesticides being teested
Wait no I'm dumb because you study something on people
Let's see how this turns out.. 😓
omg my assignment isn't tellinhg me what I did wrong/right like it usually does
I'm cooked
Okay thanks nonetheless
.close
final saddleBOT
final saddleBOT
Available help channel!
tall vector
I found y' but don't know what to do next
final saddleBOT
I found y' but don't know what to do next
tired walrus
can you write an expression for the distance from the particle to the origin in terms of x and y (or even just x)?
it'll be convenient to give it a name too, so call it r
tall vector
So just sqrt[x^2+y^2] ?
soft zealotBOT
Ann
tired walrus
you have that dx/dt = 3
and you want dr/dt when x=4.
tall vector
Oh ok
tired walrus
do you see how to continue now?
tall vector
I just plug in x and x' and solve for r' right?
tired walrus
$y = \sqrt{x}$ and therefore $y^2 = x$.
soft zealotBOT
Ann
tired walrus
if you wanted me to be more bureaucratic i would have written $$r = \sqrt{x^2 + (\sqrt{x})^2}$$ perhaps.
soft zealotBOT
Ann
tired walrus
your r' is wrong
chain rule misapplied on the root!
also if we're doing time derivatives then might as well write them with over-dots instead of primes
you'll have $$\dot{r} = \frac12 (x^2+x)^{-1/2} (2x+1) \dot{x}$$
soft zealotBOT
Ann
tired walrus
or in more familiar notation
$\dv{r}{t} = \frac{2x+1}{2\sqrt{x^2+x}} \cdot \dv{x}{t}$
soft zealotBOT
Ann
tired walrus
@tall vector do you understand
tall vector
Yes
Why shouldn't I use chain rule for something like this, but instead distribute
tired walrus
Why shouldn't I use chain rule for something like this, but instead distribute
bc if you dont distribute then you have to use both chain AND product
and you used the fool's product rule
also, unrelatedly,
is this (a) $\frac{1}{3}\pi$ or is it (b) $\frac{1}{3\pi}$?
soft zealotBOT
Ann
tall vector
It's a 🥲
tired walrus
ok well dont fucking write it in such a way that the pi looks like it's in the denom mate
either write $\frac13 \cdot \pi$ or something, or $\frac\pi3$ or just pay 30% more attention to how you place symbols on the page with your pen(cil)
soft zealotBOT
Ann
tired walrus
anyway
(3.25h)^2 * h
if you dont distribute this
youre gonna need the product rule
AND the (...)^2 part will require the chain rule
its gonna be a royal mess
and your life will be difficult and you will question your life choices up to that moment
tall vector
Okay so basically I should distribute whenever I can and use chain rule/or just take derivative of the variable if everything else is a constant?
tired walrus
i am not going to give you any sort of directive to "distribute under xyz conditions"
but what you SHOULD do is SIMPLIFY your functions BEFORE you take their derivatives, IF such an opportunity presents itself.
final saddleBOT
final saddleBOT
Available help channel!
boreal smelt
final saddleBOT
boreal smelt
i am blank. How do i start?
tired walrus
let the members of the arithmetic progression be a, a+d, a+2d, a+3d
then
a+2, a+d+7, a+2d+9, a+3d+5
are the original x_i and thus form a GP
it'll take a bit of crunch but you can find a and d from here surely
that or try doing a similar thing but starting from the gp
boreal smelt
then
a+2, a+d+7, a+2d+9, a+3d+5
are the original x_i and thus form a GP
won't they be
a-2, a+d-7....?
tired walrus
no
umbral hamlet
i would definitely start from the geometric representation, there's a lot more structure
tired walrus
you subtract 2, 7, 9, 5 FROM the gp to make ap
tired walrus
i would definitely start from the geometric representation, there's a lot more s...
ig so
maybe it's less ugly that way
umbral hamlet
yes, it is
boreal smelt
then
a+2, a+d+7, a+2d+9, a+3d+5
are the original x_i and thus form a GP
so you mean this is the gp
boreal smelt
let the members of the arithmetic progression be a, a+d, a+2d, a+3d
and this is the ap
?
boreal smelt
b-2, br-7, br^2-9, br^3-5 form an AP
so now my job is to find b and r?
tired walrus
yes
soft heart
tired walrus
i don't
boreal smelt
i got an eqn $br^2 - 2br + b +3=0$
soft zealotBOT
Prathmesh
tired walrus
there are plenty of questions which i skip
tired walrus
i got an eqn $br^2 - 2br + b +3=0$
you should have two eqs
soft zealotBOT
Prathmesh
umbral hamlet
great
now we need to solve these for b and r
normally that would be difficult, but these equations are structured very nicely
boreal smelt
$r=2$
soft zealotBOT
Prathmesh
umbral hamlet
yup
boreal smelt
$b=-3$
soft zealotBOT
Prathmesh
umbral hamlet
looks good
boreal smelt
now we got the terms
umbral hamlet
the geometric representation is very handy now
because we want a product of the original terms, which is easy to write in terms of b and r
final saddleBOT
boreal smelt
.reopen
final saddleBOT
✅ Original question: #help-36 message
boreal smelt
whole halo
you should open a new channel instead of reopening this one
winter halo
First write the general kth term of the sequence
boreal smelt
ohhok
whole halo
also you should tell people if the dot there means * or a period
winter halo
First write the general kth term of the sequence
.
boreal smelt
First write the general kth term of the sequence
$\sum_{n=1}^{n=\infty} \frac{n}{n^4+3n^2+4}$
soft zealotBOT
Prathmesh
green pebble
+n^2 - n^2
winter halo
$(n^2-n+2)(n^2+n+2)$
soft zealotBOT
DragunSlayar
boreal smelt
**DragunSlayar**
how did you do that?
soft zealotBOT
DragunSlayar
boreal smelt
yea
winter halo
$(n^2+2)^2-n^2$
soft zealotBOT
DragunSlayar
winter halo
$a^2-b^2 = (a+b)(a-b)$
soft zealotBOT
DragunSlayar
boreal smelt
yea thanks
winter halo
Np
restive sinew
final saddleBOT
restive sinew
What would be nth term?
pulsar axle
frail moat
show some work
final saddleBOT
@restive sinew Has your question been resolved?
restive sinew
Multiplication
frail moat
Multiplication
of?
frail moat
np
restive sinew
A/(n-1)+B/2n+C/(2n+1)
A(2n)(2n+1)+B(2n-1)(2n+1)+C(2n-1)(2n)
So for n=0
B=-1
WHEN N=1/2
A=2
when n=-1/2
C=2
2/(2n-1)-1/2(2n)+2/(2n+1)
frail moat
!done
final saddleBOT
If you are done with this channel, please mark your problem as solved by typing .close
final saddleBOT
@restive sinew Has your question been resolved?
inner sundial
which one of these lines is steeper?
final saddleBOT
which one of these lines is steeper?
inner sundial
i think this is a problem where my english isnt too good, or something like that. because I have no idea which of these 2 lines is steeper, and everyone around me diminishes this as super simple
but its been giving me a lot of trouble in physics (this question is about isothermal and adiabatic processes in actuality)
zealous stump
left is steeper? maybe show ur original question
inner sundial
the actual question is just physics
at the end of all of that, it comes down to which of the two lines is steeper
zealous stump
well left cuz it goes down more while going to the right less (rise/run)
bold turtle
i think this is a problem where my english isnt too good, or something like that...
If this is a language issue, in what language are you working?
inner sundial
If this is a language issue, in what language are you working?
hindi
thats what i speak primarily nowadays, but I've been taught the concept in english
I'm fine with english, but I can't for the life of me ever determine which graph is steeper
its a recurring issue
bold turtle
"steeper" means "more vertical-like"
inner sundial
"steeper" means "more vertical-like"
ahhh, alright. so that would be the leftward one right?
lime sun
I'm fine with english, but I can't for the life of me ever determine which graph...
lemme be simple'
bold turtle
ye
bold turtle
Speak hindi, if you like
You shouldn't have to compel him to
lime sun
and when u imagin that]
frail moat
You shouldn't have to compel him to
If it helps to understand
bold turtle
Then you speak Hindi to him if you have an explanation that works
[good edit though]
inner sundial
haha, all good
bold turtle
This is a good diagrammatic example (though the slopes having numbers is a little misleading, since there's not much information here to determine its value in the first place)
But the visual aid should be enough to get the idea
inner sundial
it is, thanks!
helps when the line is straight,its a bit annoying when its curves like decay curves and what not, but I think I got the crux of it now
thanks again
final saddleBOT
If you are done with this channel, please mark your problem as solved by typing .close
bold turtle
-# Good bot, have a cookie
inner sundial
.close
final saddleBOT
final saddleBOT
Available help channel!
velvet pivot
is it wrong to divide both sides by 5 here?
final saddleBOT
is it wrong to divide both sides by 5 here?
runic needle
no it's never wrong to do that
velvet pivot
But my answer is different from my lesson
velvet pivot
They factored x out of it instead of dividing by 5
bold turtle
is it wrong to divide both sides by 5 here?
You'd still have to factor x out here anyways
Actually complete the question here and you'll end up with the same answers
velvet pivot
You'd still have to factor x out here anyways
oh you're right
velvet pivot
Actually _complete_ the question here and you'll end up with the same answers
even with my x=-6/5 result?
bold turtle
how did you conclude that
velvet pivot
_how did you conclude that_
Well i completed the square i think
half of b squared added to both sides?
and then that's just a perfect square of 6/5 because 36/25
bold turtle
But from what
velvet pivot
But from what
i calculated it in my head, i don't know how i got this
bold turtle
is it wrong to divide both sides by 5 here?
But I mean what line here are you completing the square of?
bold turtle
If you were to complete the square here (though again you could just factorise the x out)
Then sure, you'd have 6/5 inside the brackets
What must you then subtract from the square to complete it?
velvet pivot
What must you then subtract from the square _to complete it_?
no then we square 6/5 which is 36/25 and add that to both sides and
i think i forgot to add it to both sides
bold turtle
ye
bold turtle
ye
bold turtle
You should do
Bearing in mind that "square-rooting both sides" is a "needs a plus-minus" operation
velvet pivot
Bearing in mind that "square-rooting both sides" is a "needs a plus-minus" opera...
Why? the RHS is in parentheses
hot hornet
Why? the RHS is in parentheses
in general, if $x^2 = y^2$, then $x = \pm y$
soft zealotBOT
Pseudo (Cat theory #1 Fan)
hot hornet
one way to see this is $x^2 = y^2 \iff x^2 - y^2 = 0 \iff (x + y)(x - y) = 0 \iff x + y = 0 \lor x - y = 0 \iff x = y \lor x = -y$
soft zealotBOT
Pseudo (Cat theory #1 Fan)
velvet pivot
in general, if $x^2 = y^2$, then $x = \pm y$
is this considered y?
bold turtle
The parenthesis isn't the problem-point; square-rooting is
hot hornet
is this considered y?
yeah so i guess in this case
you have $a = x + \frac 65$ and $b = \frac 65$, and $a^2 = b^2$
velvet pivot
Okay ty
bold turtle
is this considered y?
That's y^2
soft zealotBOT
Pseudo (Cat theory #1 Fan)
velvet pivot
That's y^2
right
velvet pivot
**Pseudo (Cat theory \#1 Fan)**
That makes sense
hot hornet
velvet pivot
is this good?
hot hornet
yep
bold turtle
ye, though you can then evaluate these
velvet pivot
Ty!
omg it's the same result!
i was wondering because i felt that finding the solutions of a line in a circle could be common in geometry and if i don't have a good conception right now (which i didn't have the best) then i would definitely forget how to do this later on in math
so i wanted to solve it myself
i had trouble with this because i thought -24/5+9/5=-15/5 and not -9/5
But i multiplied the 3 by 3 instead of 5
i understand it now, thanks a lot! 🩷
.close
final saddleBOT
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Available help channel!
void vessel
how are electron movements instantaneous (the exitations etc..)
since it's a process that... certainly don't seem like happening instantaneously.
and what is it that gives off the photons? the movement of an electron or the change of shell of an electron
final saddleBOT
how are electron movements instantaneous (the exitations etc..)
since it's a pr...
green grail
how are electron movements instantaneous (the exitations etc..)
since it's a pr...
i think youll get some better answers in the physics discord
should be in #old-network but yea idk if youll get the answer u want here
final saddleBOT
@void vessel Has your question been resolved?
void vessel
.close
final saddleBOT
final saddleBOT
Available help channel!
velvet pivot
Can I get some help with this please? My answer doesn't match any of the solutions, should I use the quadratic formula?
final saddleBOT
Can I get some help with this please? My answer doesn't match any of the solutio...
velvet pivot
This is my work
loud sundial
This is my work
what happened here
empty bridge
This is my work
when you have -8, it's like -1 times that thing
so -(a+b) is -1*a + -1*b
it's -x**2 + 4x -4
spiral stump
Can I get some help with this please? My answer doesn't match any of the solutio...
I can help let me solve it give me a second
velvet pivot
what happened here
first i divided both sides by 2, and then i took out a gcf of -
And then I took b/2^2 and added it to both sides
empty bridge
Can I get some help with this please? My answer doesn't match any of the solutio...
also can't you just substitue each value in?
velvet pivot
it's -x**2 + 4x -4
can you use the bot thing i can't visualize this easily in my head
empty bridge
forget it
velvet pivot
also can't you just substitue each value in?
Into the quadratic formula?
spiral stump
can you use the bot thing i can't visualize this easily in my head
Can you tell me your steps?
velvet pivot
i could but i wanted to test first if i could try other methods because the quadratic formula is a bit tedious
velvet pivot
I can help let me solve it give me a second
Okay thanks
velvet pivot
Can you tell me your steps?
First, the given is this system of equations, I set the first equation which is y equal to a quadratic set to -4x+16 equals that because in the 2nd expression that's what y is equal to
Then I set the new expression equal to zero to have it in standard form by adding 4x to both sides and subtracting 16 from both sides
spiral stump
So first thing you did is solve the system then set them equal?
velvet pivot
So first thing you did is solve the system then set them equal?
solving it is just finding the x and y values that satisfy both equations
to do that you could for example do what i did, i think
empty bridge
solving it is just finding the x and y values that satisfy both equations
I've never seen anything like this
spiral stump
The answer should be (2,8)
empty bridge
interesting
velvet pivot
interesting
i don't remember taking systems of linear-quadratics before but this is in algebra 1b in my school
velvet pivot
The answer should be (2,8)
can you tell me what i did wrong? i'm not sure
empty bridge
can you tell me what i did wrong? i'm not sure
so this is just solving -2x**2 + 8x - 8 = 0?
lemme try to do it
2 sec
velvet pivot
Okay
well once you get the x value, you can input it to find out the y value for each solution to x or something like that
spiral stump
can you tell me what i did wrong? i'm not sure
Your mistake happens in the first step of completing the square you dropped the negative sign
velvet pivot
The whole point is to figure out where on the graph the linear line and quadratic curve intersect
empty bridge
can you tell me what i did wrong? i'm not sure
there is no negative in front of the brackets
that's ur mistake
velvet pivot
Your mistake happens in the first step of completing the square you dropped the ...
I'm supposed to not?
spiral stump
I'm supposed to not?
Yes
velvet pivot
that's ur mistake
i divided both sides by -1
empty bridge
i divided both sides by -1
uh
candid pulsar
it shouldve been x^2 - 4x + 4 = 0
empty bridge
that's not what you did tho
empty bridge
it shouldve been x^2 - 4x + 4 = 0
yes
velvet pivot
that's not what you did tho
i also did other steps
candid pulsar
whered you get that extra 8 on the rhs
velvet pivot
it shouldve been x^2 - 4x + 4 = 0
Yea
frail moat
can you tell me what i did wrong? i'm not sure
where did you bring 8 after taking (-1) common?
spiral stump
I'm supposed to not?
Actually you are supposed to keep and use the negative sign but you are NOT allowed to drop it
velvet pivot
it shouldve been x^2 - 4x + 4 = 0
i got this first but then i realized that b/2^2 is 4 and that if i transfer 4 from the original expression to the 0 side and add 4 to both sides i get this so i just deleted the zero and put 8
velvet pivot
Actually you are supposed to keep and use the negative sign but you are NOT allo...
i see, i was wondering because in the actual graph the curve would have to point downwards because -a in standard form means it points down
empty bridge
can you tell me what i did wrong? i'm not sure
this quadratic only has one solution doesnt it?
i'm solving it and I only found one solution
velvet pivot
this quadratic only has one solution doesnt it?
Yea
empty bridge
x = -2
velvet pivot
Ah
empty bridge
in my multiple choice there's no -2 for the x though
since u have ur x value u can get ur y value and u should find 2,8
spiral stump
i see, i was wondering because in the actual graph the curve would have to point...
Exactly
velvet pivot
if it's 2 then if we take any one of the original expressions and input 2 into x you can find that for example y=-4(2)+16 = 8
candid pulsar
i got this first but then i realized that b/2^2 is 4 and that if i transfer 4 fr...
hmm..
what does (b/2)^2 = 4 have to do with your actions
velvet pivot
since u have ur x value u can get ur y value and u should find 2,8
Yeah
how can i get 2 though?
empty bridge
in my multiple choice there's no -2 for the x though
to be honest it made sens that the quadratic only has one solution otherwise the system wouldnt work
empty bridge
how can i get 2 though?
i'll show u$
velvet pivot
hmm..
what does (b/2)^2 = 4 have to do with your actions
Well i don't see a value that satisfies a times c equals a+c=b
velvet pivot
to be honest it made sens that the quadratic only has one solution otherwise the...
That makes sense
candid pulsar
Well i don't see a value that satisfies a times c equals a+c=b
are you trying to split the middle term
oh there
never mind
velvet pivot
are you trying to split the middle term
There's a formula in my class that you can use to 'complete the square'
frail moat
there are two soln for original quadratic
velvet pivot
geometrically
candid pulsar
There's a formula in my class that you can use to 'complete the square'
yeah that
velvet pivot
Yeah
empty bridge
there are two soln for original quadratic
no only one otherwise the system doesnt work
graph it
frail moat
there are two soln for original quadratic
but you have to consider common
empty bridge
it crosses the x axis only one time
velvet pivot
no only one otherwise the system doesnt work
you can also use the discriminant to count the number of solutions and what type
candid pulsar
no only one otherwise the system doesnt work
were talking in general
empty bridge
Ah
empty bridge
if the quadratic only has one solution the discriminant HAS to be 0
velvet pivot
if the quadratic only has one solution the discriminant HAS to be 0
Right
empty bridge
in the vertex form of the quadratic
velvet pivot
Does that mean it's neither complex nor real or are 0 and above 0 both real and only below 0 imaginary?
empty bridge
when the discriminant is 0, the root of the discriminant cancels out,
velvet pivot
when the discriminant is 0, the root of the discriminant cancels out,
Yea because sqrt 0 is just 0
empty bridge
the vertex in the x axis of a quadratic is -b/2a
if u look at the quadratic formula
doesnt it look oddly similar
so the root of the discriminant has to be 0
empty bridge
Does that mean it's neither complex nor real or are 0 and above 0 both real and ...
discriminant above 0: 2 solutions
discriminant=0, one solution
velvet pivot
discriminant above 0: 2 solutions
and 0 is only one solution
and below 0 is 2 complex
empty bridge
discriminant strictly less than zero= 2 complex solutions
empty bridge
np
velvet pivot
Thanks for the help!
that's not how they did it XD
empty bridge
that's not how they did it XD
both ways are right
empty bridge
but their way is definitely better i dont know why i didnt think of it
velvet pivot
but their way is definitely better i dont know why i didnt think of it
i was just trying whatever works first
and whatever is simpler and more elegant intuitively
empty bridge
we couldve indeed literally just set both equations equal
velvet pivot
we couldve indeed literally just set both equations equal
that's what i did
but i didn't group them at the last part
instead i found the perfect square
which is also a good method
empty bridge
when you're completing the square don't try to simplify the expression in other ways just divide by the leading coefficient
like say u have 8 x squared + 5x - 8
velvet pivot
when you're completing the square don't try to simplify the expression in other ...
but earlier someone said you can't divide when it's -?
empty bridge
divide everything by the leading coefficient which is 8
empty bridge
but earlier someone said you can't divide when it's -?
why not?
empty bridge
and?
velvet pivot
idk
empty bridge
I see what ur talking about
velvet pivot
my answer clearly was wrong xd
velvet pivot
I see what ur talking about
so you can't divide both sides by -?
empty bridge
because a quadratic with a negative a coefficient points downwards
we're solving an equation here
empty bridge
so you can't divide both sides by -?
yes you can
empty bridge
we're solving an equation here
say u have 2 ligns of an equation
the lines are equivalent but not equal
-2x**2 + 6x is not the same as x = whatever
we're solving an equation here
velvet pivot
the lines are equivalent but not equal
that's not the same thing?
empty bridge
we're not looking for two equalities here we just want a value for x
empty bridge
yea, like those two lines arent the same thing but they're equivalent
we're looking for an equivalence
velvet pivot
yea, like those two lines arent the same thing but they're equivalent
Yea i get what you are talking about
empty bridge
to be honest if u grasp this concept you're good
velvet pivot
i suppose that if you substitute this then it won't fit both equations
empty bridge
it's often the understanding part that's hard
empty bridge
i suppose that if you substitute this then it won't fit both equations
obviously not
most of the time math problems usually have very easy answers
like 2
velvet pivot
to be honest if u grasp this concept you're good
Yeah because I learned real life examples of this that are really intuitive to me already, such as when pennies and dollars have the same thing
empty bridge
most of the time although
ur 4 + or minus root 8 isnt in any of the solution and just by looking at it
it wouldnt solve either
velvet pivot
it's often the understanding part that's hard
because you need to be able to generalize for everything instead of just knowing the answer
empty bridge
because you need to be able to generalize for everything instead of just knowing...
Yes
velvet pivot
like 2
i've had answers similar to the sqrt one in the image that are true
in my class
but yeah they're mostly answers taht are basic
empty bridge
in my classes since we are not allowed calculators during exams our professors usually put equations that lead to integer answers
usually
they trick you sometimes but still
gotta keep an open mind
velvet pivot
in my classes since we are not allowed calculators during exams our professors u...
Yeah because some questions aren't tedious enough for a calculator
empty bridge
when I get a weird answer I always double check
velvet pivot
when I get a weird answer I always double check
and for systems of equations you can graph the result
empty bridge
in your head?
velvet pivot
there are so many methods to check and backtrack
empty bridge
ngl that's a bit hard
empty bridge
i dont think you would be allowed a calculator in an exam with something like this
velvet pivot
because in systems of equations, especially irl examples, the answer isn't always an integer and so it's hard to approximate it
velvet pivot
i dont think you would be allowed a calculator in an exam with something like th...
i didn't use one for this one i think. but i did use one for this
because intuitively i don't know that 157x157 equals 24,649
empty bridge
i didn't use one for this one i think. but i did use one for this
yea scrap completing the square for massive equations
too much room for error
velvet pivot
too much room for error
Yea
empty bridge
also your calculator can solve polynomial equations on its own
u just write out the coefficients and it can do it itself
I use that in physics exams to avoid the headache
velvet pivot
too much room for error
i wrote this in self evaluation, it's like what you said
empty bridge
because there's no way I'm doing 789504 squared
empty bridge
i wrote this in self evaluation, it's like what you said
do you have a physical calculator
velvet pivot
also your calculator can solve polynomial equations on its own
Yeah but i'm still in basic algebra where i need to do it on my own first to understand why it works and how it works
velvet pivot
do you have a physical calculator
app
empty bridge
alr
velvet pivot
Can i send you a friend request on discord!
empty bridge
Sure
velvet pivot
Yay
frail moat
we can move to discussion, if q done
velvet pivot
Thanks everyone for helping me with this question, it wasn't easy
calm ermine
we can move to discussion, if q done
😭(||yes||)
velvet pivot
we can move to discussion, if q done
.close
gentle zephyr
final saddleBOT
soft zealotBOT
Renato
gentle zephyr
basically since 70 = 2x5x7, then 5 does not divide this number 24a^25.... but 2 and 7 does
call this number 24a^25 -2a^19 -2a = X, then
2 | X, 7 | X, but 5 does not divide X
24a^25 -2a^19 -2a = 0 (mod 7)
3a^25 -2a^19 -2a = 0 (mod 7)
3a^25 + 5a^19 + 5a = 0 (mod 7)
a^25 + 25a^19 + 25a = 0 (mod 7)
a^25 + 4a^19 + 4a = 0 (mod 7)
so one possible solution is a = 0 (mod 7)
i think
cursive bough
yep that is true. I would use fermat's little theorem here
simplifies things perfectly actually
cursive bough
(also notice that you are assuming that 7 does not divide a here, it's important)
gentle zephyr
(also notice that you are assuming that 7 does not divide a here, it's important...
a is a integer value from 0 to 6
only case that this shit is gcd(a,7) ≠ 1 is when a = 0
hasty mist
You haven’t worked mod 5 yet
gentle zephyr
your point?
hasty mist
hasty mist
if n = 7, then gcd(7,7) = 7
What is your confusion with this?
gentle zephyr
the confusion was that I didn't noticed that a is in-between 0 and 6
the multiples of 7 are not considered as they count as 0 under mod 7
wait a second I am getting confused
hasty mist
Okay, so do you agree that the only way to get divisibity by 7 is 7|a?
gentle zephyr
Okay, so do you agree that the only way to get divisibity by 7 is 7|a?
ok sure
after simplifying with fermat a^19 = a(a^6)^2 = a = 0 (mod 7)
a^25 = a(a^6)^4 = a = 0 (mod 7)
hasty mist
Yes. So the whole thing becomes 2a(5-1-1) = 2a(3) = 6a mod 7
Hence, we need 7 | a.
That's one of the conditions.
Now you can consider the condition for modulo 5. We need that 5 does not divide 2a(12a^25 - 2a^19 -2a).
gentle zephyr
Yes. So the whole thing becomes 2a(5-1-1) = 2a(3) = 6a mod 7
how?
dont skip steps please
hasty mist
\[24a^{25} - 2a^{19} - 2a == 2a(5a^{24} - a^{18} - 1) == 2a(5\cd1 - 1 -1) \pmod7\]
wooden bridge
Need an help in French !
soft zealotBOT
Cooly
hasty mist
Need an help in French !
Please see #❓how-to-get-help
hasty mist
Hm?
gentle zephyr
Yes. So the whole thing becomes 2a(5-1-1) = 2a(3) = 6a mod 7
6a = 0 (mod 7) => 7 | a?
gentle zephyr
Now you can consider the condition for modulo 5. We need that 5 does *not* divid...
we can find the ones that work and exclude those
hasty mist
6a = 0 (mod 7) => 7 | a?
Yeah, 6 is invertible in mod 7.
hasty mist
we can find the ones that work and exclude those
Yes.
gentle zephyr
Yeah, 6 is invertible in mod 7.
gcd(6,7) = gcd(6,1) = gcd(0,1) = 1
hasty mist
I mean sure. 6 and 7 are coprume.
gentle zephyr
since gcd(6,7) = 1, multiplicative inverse of 6 (mod 7) exists
24a^25 -2a^19 -2a = 0 (mod 5)
hasty mist
since gcd(6,7) = 1, multiplicative inverse of 6 (mod 7) exists
What is it then?
gentle zephyr
36 = 35 + 1
6x6 = 7x5 + 1
inverse of 6(mod 5) is 6
24a^25 -2a^19 -2a = 0 (mod 5)
hasty mist
Okay sure
gentle zephyr
gcd(a,5) = ?
if a ≠ 0 then gcd(a,5) = 1
a^25 = a(a^4)^6 = a (mod 5)
a^19 = a^3 × (a^4)^4 = a^3 (mod 5)
24a^25 -2a^19 -2a = 0 (mod 5)
4a + 3a^3 + 3a = 0 (mod 5)
7a + 3a^3 = 0 (mod 5)
2a + 3a^3 = 0 (mod 5)
@hasty mist i might need some helping hand
a = 0, a = 1, a = 4, I just tried every number a from 0 to 4
hasty mist
X = 2a(2a^24 - a^18 - 1) (mod 5)
= 2a(2 x 1 - a^2 - 1) (mod 5)
= 2a(1 - a^2) (mod 5)
Isn't that right?
hasty mist
```tex
X = 2a(2a^24 - a^18 - 1) (mod 5)
= 2a(2 x 1 - a^2 - 1) (mod 5)
=...
If you check for a = 1,2,3,4 here I think you get that a cannot be 1 or 4.
hasty mist
a = 0, a = 1, a = 4, I just tried every number a from 0 to 4
Yeah, so a == 2 or a == 3 mod 5
Now you can use CRT to combine those conditions.
\[
\t{System }1 = \begin{cases} a == 0 \pmod 7\\ a == 2 \pmod5\end{cases}
\]
\[
\t{System }2 = \begin{cases} a == 0 \pmod 7\\ a == 3 \pmod5\end{cases}
\]
Which is the same thing anyway..
gentle zephyr
what?
hasty mist
sorry
hasty mist
when did the china theorem came into play
From the first condition, we have that a == 0 mod 7
Yes?
gentle zephyr
sure
hasty mist
And in order for 5 not to divide X, you need a == 2 or 3 mod 5
So you can combine those conditions with CRT.
gentle zephyr
can't we solve just one system
why we need 2
because its an OR not an AND
i guess so
hasty mist
I don't know why I said that. I'm under some influence of opioids due to being hospitalised. You need to solve both systems of equations.
\[
\t{System }1 = \begin{cases} a == 0 \pmod 7\\ a == 2 \pmod5\end{cases}
\]
\[
\t{System }2 = \begin{cases} a == 0 \pmod 7\\ a == 3 \pmod5\end{cases}
\]
Cooly
gentle zephyr
I don't know why I said that. I'm under some influence of opioids due to being h...
i see, though you are still functioning very well
i like dat
hasty mist
why we need 2
The two congruences are equibalent in isolation but yhey yield different solutions when combined with a == 0 mod 7
gentle zephyr
Mi yi = ci ( mod mi )
5 y1 = 0 (mod 7)
50 = 49 + 1
y1 = 0 (mod 7)
7 y2 = 2 (mod 5)
21 = 20 + 1
y2 = 6 (mod 5)
y2 = 1 (mod 5)
y1 = 0, y2 = 1
xi = Mi yi
x1 = 5 × 0 = 0
x2 = 7 x 1 = 7
a = 0 + 7 = 7 (mod 35)
gentle zephyr
```tex
\[
\t{System }1 = \begin{cases} a == 0 \pmod 7\\ a == 2 \pmod5\end{cases}...
the solution to the first system is just a = 2 (mod 5)?
smoky egret
is 2\equiv 0 mod 7?
gentle zephyr
no
smoky egret
the solution to the first system is just a = 2 (mod 5)?
so is this a correct solution to the first system?
if the solution is a\equiv 2 mod 5 then a=2 should be a solution since 2\equiv 2 mod 5 right?
and if 2 is a solution to the first system then 2\equiv 0 mod 7 but thats not the case
smoky egret
**Cooly**
for systems like this with small moduli (and a small number of equations in the system) the fastest way is to probably manually find a solution
so first you want a number a that satisfies a\equiv 0 mod 7
whats the smallest natural number that satisfies this?
smoky egret
by RRT you mean chinese remainder theorem ig, you can do that np but its faster for you to manually find the solutions
like in the exam you need this extra time that you can save
gentle zephyr
a = 0 + 7 = 7 (mod 35)
is just this
smoky egret
nice
gentle zephyr
i was taking modulo 5 instead of mod 5x7
gentle zephyr
Mi yi = ci (mod mi)
M1 = 5, M2 = 7
5 y1 = 0 (mod 7)
15 = 14 + 1
y1 = 0 (mod 7)
7 y2 = 3 (mod 5)
21 = 20 + 1
y2 = 9 (mod 5)
y2 = 4 (mod 5)
xi = Mi yi
x1 = 5 × 0 = 0
x2 = 7 × 4 = 28
a = 0 + 28 = 28 (mod 35)
that should do the trick @smoky egret @hasty mist
smoky egret
Mi yi = ci (mod mi)
M1 = 5, M2 = 7
5 y1 = 0 (mod 7)
15 = 14 + 1
y1 = 0 (mod 7)
7...
nice job
hasty mist
that should do the trick <@769420534537912331> <@1421147725335891981>
Well done.
gentle zephyr
I don't know why I said that. I'm under some influence of opioids due to being h...
hope your recovery goes well
fringe quarry
I’m a little confused, if f(0)=0 how does that immediately imply C=0?
tired walrus
ln|0+1|=?
worldly spruce
so fucken cool
tired walrus
also where are the dx's
worldly spruce
interesting
fringe quarry
yeah but it’s not working for the problem I’m on rn
tired walrus
is this the tik tok guys book
no it has too many symbols per page to be targeted to tiktokers
fringe quarry
can’t really do a=0 😔
rocky tusk
no it has too many symbols per page to be targeted to tiktokers
wrong
rocky tusk
book of integrals is fire
nah this shit is ass
tired walrus
how did pi happen
rocky tusk
typesetting lowkey hurts my eyes
fringe quarry
how did pi happen
tired walrus
can’t really do a=0 😔
this looks like a completely different function
fringe quarry
correct me if I am wrong
fringe quarry
this looks like a completely different function
It’s a different question
I’m just confused cos like
tired walrus
so you put some other value of alpha that works
fringe quarry
can I make a= anything?
tired walrus
0 isn't sacred
fringe quarry
well yeah
but then I also get f(1)=C which doesn’t really help I think
the answer is 0 but I don’t see how I can get that
tired walrus
honestly maybe feynman technique ain't very good here
how about x=e^u on the original integral
fringe quarry
Hmmm I’ll try that
bold turtle
Slight NB about the notation here:
Once you've integrated that dx, the function is in alpha only
So the derivative outside the integral, i.e. of f(alpha), is a full derivative (not partial)
Also, you mentioned alpha := 0 doesn't lead you anywhere
Sure, not at the end, because of the log
fringe quarry
mmm yeah I see that vision
bold turtle
But what about where you defined f in the first place?
fringe quarry
f still has lnx on the top so 0 still doesn’t work I think
)
final saddleBOT
final saddleBOT
Available help channel!
severe lava
final saddleBOT
severe lava
there *is* an asymptote here, i believe, but i just want to confirm <:catthumbsu...
ok so my maths teacher and i quote believes its non sense
and he told me to rewatch the youtube video
1 second ill show u what way he does it
and he's asking me why the numerator and denominator are the roots or whatever
and he thinks
its a conwindence it works
and he asked me why it works
tribal ocean
its not
severe lava
ikr
tribal ocean
wait which way did you do it
sour zealot
is your teacher... chatgpt? we usually advise against using AI here, as it can lead to misinformation
tribal ocean
this one or prev
severe lava
+this is the way i self thought from yt
tribal ocean
that's how i learnt it lol
severe lava
is your teacher... chatgpt? we usually advise against using AI here, as it can l...
no no, i just used chatgpt cause idk latex
tribal ocean
in school
severe lava
and its easier for your guys
tribal ocean
+this is the way i self thought from yt
this is exactly how i learned it im p sure
it's very easy to prove why it works
sour zealot
that's how i learnt it lol
imma dip, best of luck to you two 
severe lava
it's very easy to prove why it works
he wants me to rewatch and prove it, do you know how ill prove it
tribal ocean
Yes it's trivial
severe lava
Yes it's trivial
should i just say its a amsyptopde and hope he understand what that means
tribal ocean
but is he asking you why your method works
severe lava
if the x is a minus
severe lava
but is he asking you why your method works
yes
tribal ocean
its proof by cases
severe lava
and he told me infront of the class that its non sense
tribal ocean
Hes dumb
severe lava
its proof by cases
yeah i told him, but he just explaion to me how it works
but ill be honest idk how it works
but it just does
tribal ocean
you should get an understanding of why then
severe lava
and i dont wanna ask ai, cause like the guy said earlier, it just might lead to mis information
tribal ocean
can you give me 5 mins i gtg
severe lava
you should get an understanding of why then
no yeah thatts why i came here to ask why
tribal ocean
I'll write the proof after
severe lava
can you give me 5 mins i gtg
yeah np
severe lava
I'll write the proof after
ok tyyysmmm
gloomy river
Your method is equivalent to your teacher's method; note that after your Step 3, if instead of doing it by cases, you multiply both sides by (x+2)^2, then you just go to your teacher's Step 4.
severe lava
Your method is equivalent to your teacher's method; note that after your Step 3,...
yeah thats one way of looking at it
but i just want to prove that
can just as easily get you
the answer
without going the long way
gloomy river
I don't think either of you is going "the long way". Doing by cases is not that easy to formalize, it might use some notions that you may have not seen (continuity, intermediate value theorem), which the parabola way avoids by using a convexity argument (have you seen what it means for a function to be convex?).
severe lava
I don't think either of you is going "the long way". Doing by cases is not that ...
i lost you
no i havent seen that lol
but ill look into it
but ill put thatt point forward, to my maths teacher, considering he probably knows what that means and it might broaden his understanding of what im trying to do
gloomy river
Yeah, no need, your teacher won't expect you to know that. The thing is, he should know that doing by cases is a formal way of doing it, so I don't think there really is a point of arguing with him; just learn his method, and he'll stop complaining.
severe lava
cause instead of sayin critical points, i said special numbers, cause i didnt know the word for it at the time
severe lava
Yeah, no need, your teacher won't expect you to know that. The thing is, he shou...
yeah thats fair enough, he asked me to rewatch the video and come back to him, he said it was 'interesting'
so 🤷♂️
im happy i have a teacher thats opened minded at least yk
tribal ocean
Yeah technically its because of halfspaces buuut I mean its just checking if each term is pos and neg and when that happens
You can argue by showing each case is handled correctly
severe lava
Yeah technically its because of halfspaces buuut I mean its just checking if eac...
yeah i need to look into that, im watching some more youtube video, and the first video never mentioned about the pos and neg thing, just told me to take a random number in between the 2 and see if it works
if it works its true
if not its false
tribal ocean
it works because they all have the same sign for each term
so checking any one of them checks everything in that interval
gloomy river
Yeah, no need, your teacher won't expect you to know that. The thing is, he shou...
But if you want a rough explanation, the intuition is that on either side of $x=-2$, the function $f(x) = \frac{-x-9}{x+2}$ is continuous, basically meaning that you can draw its graph without lifting your pen from the paper. This means that if $f(x)$ were to change sign on either on the intervals $(-\infty,-2)$, $(-2,\infty)$, it would have to cross the $x$-axis (it would have to equal 0). Since the only spot where $f(x)=0$ is at $x=-9$, then it doesn't change sign on teh intervals $(-\infty,-9)$,$(-9,-2)$, and $(-2,\infty)$.
soft zealotBOT
Feltheshovel
tribal ocean
yes continuity ftw
severe lava
But if you want a rough explanation, the intuition is that on either side of $x=...
ok ty 🙂
ty so much guys for ur help
i appriciate it so so much
gloomy river
Also, don't always look at things negatively; he may have feigned to doubt you so you would try to understand why your own method works. Even if that was not his intention, you must admit that you learned more from this than if he had simply told you that you were correct.
severe lava
Also, don't always look at things negatively; he may have feigned to doubt you s...
that exactly how im looking at it lol, ill be honest, i didnt have a clue how it worked
and it was nice learning about it
now i like properly understand it
.close
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solar crest
need latex help
how do I make my background white?
final saddleBOT
need latex help
how do I make my background white?
plucky rover
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shi
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Sniped
versed crater
Yes
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Yeah
versed crater
Move theee
solar crest
yes to which
plucky rover
The first
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latent dragon
,tc color white or somehting like that
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latent dragon
oops.
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The latex people hangout around there
solar crest
🥀
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gentle zephyr
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Renato
gentle zephyr
1 + 1 + a + 2 + a + 1 - 1 ≠ 0
2a + 4 ≠ 0
a ≠ -2
w^3 = e^{2ki.pi/3} , k ∈ {0,1,2}
storm haven
w^3 = e^{2ki.pi/3} , k ∈ {0,1,2}
Isn't this w, not w^3?
storm haven
Yes lol you wrote w^3 for the right side equation in your previous message
storm haven
Well, notice that no matter what w (unless it is 1) is, {w, w^2, w^3} is the 3 roots of unity
storm haven
so you saying w^3 = e^{2kpi × i}
w not w^3
gentle zephyr
so w^3 is?