you give yourself not nearly enough credit
#help-23
neon vapor
cloud hound
If you have any feedback I'm very willing to hear though 👌
cloud hound
you give yourself not nearly enough credit
Yeah grading my classmates work I've seen some horrors 💀 😅
Proofs with no words
All symbols 💀
neon vapor
i see both that and proofs with only words
cloud hound
I'm sure you would love some of my proofs from freshman year where I get distracted in the middle and have to come back later to finish
No no that makes sense, I've never heard of structural induction tbh so I was just making up what I thought might be right
safe radishBOT
@cloud hound Has your question been resolved?
cloud hound
you give yourself not nearly enough credit
Oh but thank you! I should say
I'll be continuing to do exercises from the book for at least a week
safe radishBOT
silk cove
x=10 mod 21
x=10 mod 22
x=10 mod 56
x=0mod37
x can be replace by 37k
looking for any kind of short approach which takes very low time
plucky elk
Find the least common multiple of 21,22,56 and express x in terms of the LCM
Then plug that expression into mod 37
silk cove
LCM of 21,22,56 is 2^3.3.7.11
i did not understand express x in terms of LCM
you meant like 37
big divide wait
1848/37 so we get
vivid bison
1848/37 so we get
Then just find lcm of those two numbers
silk cove
i got 1848=37.49+35
vivid bison
How's that useful?
vivid bison
Do yk how to find lcm?
silk cove
35=-2 mod 37
silk cove
Do yk how to find lcm?
1848 is LCM
vivid bison
Now find lcm of 1848 and 37
silk cove
Then plug that expression into mod 37
please check my work
plucky elk
please check my work
Check what exactly
quasi bison
x-10 is a multiple of LCM(21, 22, 56) and x-10 ≡ -10 (mod 37)
so we should be looking for the least value of x-10 in light of this, i think
@silk cove
silk cove
x-10 ≡ -10 (mod 37)
ohh 1848k+10 mod 37
so we need to check by putting k=1,2,3,4
so on
k=5 works
ohh simple
.close
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strong pike
Question and book solution
safe radishBOT
Question and book solution
strong pike
Question and book solution
I added both amounts then applied discounts. Seems right as per information given in question.
Am I wrong?
quasi bison
you're applying both discounts to both bills at once
in reality the 9800 bill is supposed to ONLY get the 2% off for 'within ten days' and the 5480 gets ONLY the 5% off for 'payment the day-of'.
these discounts don't stack in any way!
safe radishBOT
@strong pike Has your question been resolved?
strong pike
these discounts don't stack in any way!
Got it. Took me a while to understand English lol. Thank you.
midnight sonnet
Can someone please walk me throught his dry run of the Dijkstra's algorithm
midnight sonnet
I mean I know how it works but I do not know how to fill this table
the table is for this graph
safe radishBOT
@midnight sonnet Has your question been resolved?
livid lodge
mhm, I'll try to explain it in the best of how I can word it. So you know the Dijkstra algorithm. It starts from a permanent point and adds a certain weight to each other point. V(T) is all the points you've crossed at step 1,2,... E(T) is the connection. F is a sort of buffer I believe, to see which points you haven't yet used. and L is the weight of each point at every step.
So at step 0 you know know that point a is 0 and all others haven't been checked so therefore they are inf
once you set that first step and check from a-b you know the weight of b
does this help? or is something in my explenation confusing?
@midnight sonnet
midnight sonnet
<@1451592087773642876>
Thank you, I will read your explanation with the graph side-by-side, if I have any queries then I'll ask again
livid lodge
Alright, feel free to ping me when you're stuck
midnight sonnet
midnight sonnet
@livid lodge This is the graph and here is the table from before, now as you can see that after we assess the distances of b and c from a, we put F as {b,c} in the table. Which are the vertices that we have checked the distance of a from. But in step #2 of this table, we are checking the distances of d and e from b. Then why do we have to include c in F too? As you can see, F in step #2 is {c,d,e}, whereas I think that it should have been {d,e}, because we are only checking b's distance from d and e, but not c.
livid lodge
F is a sort of buffer, it stores the points not checked. So in Step 0 you have a in F, so that's the on you check in that step and fill in all the neighbours of a. These are 'temporary points'(idk how your professor explained Dijkstra, that's how he explained it to me).
So next step, you have {b,c} in your buffer, so you see the weight of each point from a. for b that's 3, c that's 4. Next step you choose the one with the lowest weight, in this case b(since b<c obv.) so we remove b from the buffer F and make it a 'permenant point'(meaning this is decided to be the shortest path from your origin point, in this case a). Then you repeat stap 0, check all the neighbours of b: d & e and fill their weight in. Once you've done that you now have the buffer F that started like this: a[origin, therefore smallest point] -> b,c [neighbours of a, with b smallers, take this one out of the buffer] -> c,d,e[neighbours of b, with c the smallest, take this one out].
This is a bit of extra info for the next step if you're still confused, text above should have everything already explained. So now we have c, we make that point permanent, and check it's neighbours, that's a and e. a has already been checked so doesn't get added to the buffer, e is already in the buffer so nothing changes. Check the weights of c to e, that's 1. Add the original weight from c + {{e,c}} and you get 5, this is FAR shorter than from a-b-e so it gets changed in the table. Now e has become the smallest point and rinse and repeat
If something is still confusing you, I'll try to visualise it more in the next explenation :) feel free to still ping me
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spare shale
"Hi everyone! I'm from the Philippines and I'm aiming to win the Division Math Celebration Quiz Bee for senior high school. I'm in Grade 11. I think the competition will focus on statistics what i would do
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"Hi everyone! I'm from the Philippines and I'm aiming to win the Division Math C...
nimble vine
what topics are you looking for?
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covert rain
$\int^1_{-2} \frac{x+1}{\sqrt{x+3}+2}dx$
flat frigateBOT
Professor Edward C. Hawthorne
safe radishBOT
$\int^1_{-2} \frac{x+1}{\sqrt{x+3}+2}dx$
covert rain
How could one understand that the best substitution was √(x+3)=t?
I tried to rationalize it since there was a root underneath, by parts...
Also it's not clear to me after the substitutions, how you break the numerator and denominator like that
covert rain
Also it's not clear to me after the substitutions, how you break the numerator a...
,rotate
,rotate
flat frigateBOT
untold magnet
no idea about why he thought about taking root x+3 as t tho
covert rain
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midnight sonnet
please give me a proof question
safe radishBOT
please give me a proof question
midnight sonnet
(either direct or contraposition or contradiction)
hard crest
prove √3 is irrational
midnight sonnet
okies
midnight sonnet
prove √3 is irrational
if a^2 = 3n, then does a = 3m for some m?
hard crest
of course it does, a = 3(n/a)
faint hornet
if a^2 = 3n, then does a = 3m for some m?
remember that each number has a unique prime factorisation
midnight sonnet
of course it does, a = 3(n/a)
what if n/a is not an integer
does that break anything
hard crest
then i suppose m wouldn't be an integer either, but what does that mean?
midnight sonnet
then i suppose m wouldn't be an integer either, but what does that mean?
i define a multiple of 3 as any integer n which can be written as n = 3k for any integer k
does k here has to strictly be an integer?
hard crest
well given that you defined it that way.... yes
midnight sonnet
well given that you defined it that way.... yes
well then n/a could not be an integer, then i cant say that a is a multiple of 3 if a^2 is
hard crest
why can n/a not be an integer?
i'm having trouble following your train of logic; could you state it in a more self-contained manner?
midnight sonnet
why can n/a not be an integer?
that is my question too
will it always be an integer
if a^2 = 3n
severe pond
if a^2 = 3n
why is n here
midnight sonnet
alright i'll just state the condition that i faced in the proof
severe pond
prove √3 is irrational
this was the question
severe pond
a^2 = 3b^2
yea so squaring would give you this instead then
midnight sonnet
now i wanna conclude a is a multiple of 3
severe pond
prove that 3 | n^2 —> 3 | n
faint hornet
(where a | b means that b is divisible by a)
midnight sonnet
a | b means b = ak?
severe pond
yes
midnight sonnet
n^2 = 3k
n = 3(k/n)
n = 3l for l = k/n
severe pond
btw in b = ak, is k always gonna be an integer
yes
severe pond
n^2 = 3k
n = 3(k/n)
n = 3l for l = k/n
how do you know that k/n is an integer
faint hornet
you wanted a proof by contradiction / contraposition question right?
you can in fact, use one of those to prove what knief is asking
so why don't you give it a go first and forget about hayley's question for now
then use this result to come back to it
severe pond
it might notbe
why?
midnight sonnet
alright well contraposition proof works
premise: 3 not | n
then n != 3k
n^2 != 9k^2
n^2 != 3(3k^2)
n^2 != 3l
hence n^2 not | 3
so 3 | n^2 means 3 | n
faint hornet
yeah, the main idea being that is 3 does not divide n, then it cannot divide n^2 either
so could you perhaps use this now to prove hayley's problem from before?
midnight sonnet
so could you perhaps use this now to prove hayley's problem from before?
do i have to include this in that proof
or is it implied (was very non obvious to me )
faint hornet
that depends on how much detail you want to put in, but it is worth knowing if it's non obvious to you
faint hornet
prove there are infinitely many primes
midnight sonnet
prove there are infinitely many primes
can i express the primes as any n belong to N such that: for all m belong to N, GCD(m,n) = n
faint hornet
if by N you mean naturals, then it is almost right; the only difference is that 1 is not prime
faint hornet
no
faint hornet
you should just think of primes as positive integers that are not 1 that are only divisible by 1 and themselves
midnight sonnet
you should just think of primes as positive integers that are not 1 that are onl...
express the primes as any n belong to N such that: for all m belong to N, GCD(m,n) = n why is this incorrect?
faint hornet
suppose you have say 3, an obvious prime
then gcd(5,3) = 1, not 3
midnight sonnet
you should just think of primes as positive integers that are not 1 that are onl...
can i express the primes as any n belong to N such that n | n and n | 1
wat now?
faint hornet
what
faint hornet
you should just think of primes as positive integers that are not 1 that are onl...
^
faint hornet
can i express the primes as any n belong to N such that n | n and n | 1
dont think about it like this at all
midnight sonnet
dont think about it like this at all
then how
faint hornet
you should just think of primes as positive integers that are not 1 that are onl...
this is enough, but also knowing that every number uniquely factors as a product of primes
okay how about we start like this
look at the question, do you think you would use contradiction or contraposition?
and why
midnight sonnet
look at the question, do you think you would use contradiction or contraposition...
contradiction, because there is no conditional here
midnight sonnet
so what do you want to assume
there are a finite number of primes
faint hornet
okay, good
so one thing we can do with this, is suppose that we have this finite list of primes p_1, ..., p_n
if we can find a prime NOT in this list, then we are finished
we're trying to look for a way to achieve this
can you think of any methods
faint hornet
this is enough, but also knowing that every number uniquely factors as a product...
with this as a hint
midnight sonnet
with this as a hint
well if for every n belong to N, there is a unique combination of product of primes, then we need an infinite amount of combination of primes to accomodate every n. That would mean that there are infinite primes
faint hornet
not quite, because we can make infinitely many numbers with even one prime! since we can just take 2^n for any n
what i am rather hinting towards is trying to find a number which is not divisible by any of these primes
midnight sonnet
not quite, because we can make infinitely many numbers with even one prime! sinc...
how can we do that?
like suppose if 3 is the only prime
midnight sonnet
you tell me!
well i think we need infinite combinations of primes to accomodate all n
thats why there are infinite primes
faint hornet
right but that is not a proof
often times in math there are things that work very unintuively
so we can't just say "i think this will happen" and then call it a proof
midnight sonnet
so we can't just say "i think this will happen" and then call it a proof
isnt it true that we need infinite combinations
faint hornet
well can you prove that
midnight sonnet
well can you prove that
its one of those conjectures or smth
if u have infinite numbers then u got infinite combinations
faint hornet
suppose our only prime was 2
then 2, 2^2, 2^3, 2^4, ..., 2^n, ... are only "combinations" of one prime
and yet you have infinitely many numbers
so this is not a valid argument for saying there are infinitely many primes
midnight sonnet
then 2, 2^2, 2^3, 2^4, ..., 2^n, ... are only "combinations" of one prime
alright then how abt i add 1 in your list and then prove that it is prime
that p_1...p_n , i add all these, and add a 1
faint hornet
that does not prove anything
midnight sonnet
okay how abt i multiply them and then add 1
faint hornet
okay, why?
midnight sonnet
okay, why?
if i can prove that the new number is also prime then there is a prime that is bigger than my original existing largest prime
i.e a contradiction
faint hornet
okay, but what if it is NOT a prime
for example, 2*7+1 = 15 is not prime
but it still proves existence of a new prime... why?
midnight sonnet
but it still proves existence of a new prime... why?
if it is composite then there must be a prime from my existing list that divides the large number formed
midnight sonnet
exactly this, so where is the contradiction
but is every composite divisible by some prime?
faint hornet
that is precisely the definition of a composite number, yes
so if the product + 1 is composite, why does a prime from our list not divide it?
midnight sonnet
so if the product + 1 is composite, why does a prime from our list not divide it...
because there is a remainder 1/p
faint hornet
sure, we can say that
midnight sonnet
ok so the list is not complete
faint hornet
more precisely it's because it would have to divide 1 also, and there is no such prime
midnight sonnet
there is a prime that is out of our list that divides our number
faint hornet
exactly
faint hornet
yep :) nice job
midnight sonnet
yep :) nice job
no
here's the thing
how do i come up with all this rocket science in the exam hall
there is less time and more pressure and problems harder than this
faint hornet
how do i come up with all this rocket science in the exam hall
okay well you know when to use contrapositive and contradiction
you know what assumption to make and you just think about why it's useful
i think, these are kind of natural proofs that you were asked so far
midnight sonnet
btw why are all composites divisible by some prime
faint hornet
btw why are all composites divisible by some prime
what is a composite number
midnight sonnet
what is a composite number
a number that divides more numbers than just 1 and itself
faint hornet
okay, and what if we keep splitting up this composite number into its divisors
midnight sonnet
okay, and what if we keep splitting up this composite number into its divisors
well you want me to say that we end up with a prime
but idk why is that the answer to this question
faint hornet
our divisors keep getting smaller and smaller
midnight sonnet
our divisors keep getting smaller and smaller
yup
faint hornet
so eventually, we need to find a number which has no divisors other than 1 and itself
which is precisely what a prime is
midnight sonnet
so eventually, we need to find a number which has no divisors other than 1 and i...
alright
.close
viscid monolith
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viscid monolith
a)
Since $a_n = \frac{a_{n-1}}{2+a_{n-1}}$ then $a_{n+1} = \frac{a_n}{2+a_n}$
$\frac{a_{n+1}}{a_n} < 1$ as $a_n >0$ it simply implies that it would be a decreasing sequence. We also know that $a_n > 0$ Hence, it is bounded by 0.
So, by the monotone convergent theorem, ${a_n}$ is convergent.
b) Since it's a decreasing monotone function, I inferred that $a_1 > a_2 > a_3 ... > a_{n-1} > a_n$. Since $\lim_{n\to \infty} [a_n] = L = \lim_{n\to \infty} [a_{n-1}]$ then
$\lim_{n\to \infty} [a_n] = \lim_{n\to \infty} \frac{a_{n-1}}{2 + a_{n-1}}$ is same as saying L = $\frac{L}{2+L}$? $\therefore$ the limit of ${a_n}$ as n tends to $\infty$ is 0.
flat frigateBOT
Maddie
viscid monolith
I tried this question and I would like to get inputs on whether I did it correctly or not?
The answer is 0 - textbook solution.
cloud hound
I can't say anything on correctness but I am wondering how you derived a_n+1 = in a)
viscid monolith
So my thinking was if the $a_n = \frac{a_{n-1}}{2+a_{n-1}}$ it would mean that at some n, the value of $a_n$ is that of the previous value. Hence, $a_{n+1}$ would be $a_n$.
flat frigateBOT
Maddie
viscid monolith
I think I worded this really really badly 
opaque fern
I tried this question and I would like to get inputs on whether I did it correct...
Yeah
Its correct
safe radishBOT
opaque fern
thankiuu ❤️
One noge
Note
You should explicitly mention why you chose to ignore -1
Otherwise, good job
viscid monolith
Sorry, what do you mean by -1?
cloud hound
So my thinking was if the $a_n = \frac{a_{n-1}}{2+a_{n-1}}$ it would mean that a...
Oh I see it's not multiplication it's separate sentences
I thought it was multiplication 😅
opaque fern
Sorry, what do you mean by -1?
Solving L = L/(2+L) gets you two solutions
viscid monolith
Oh!!! Yes yes, thank you so much for your help!
keen tulip
you're welcome
magic junco
.solved
glossy bear
Can someone check if this proof is right?
safe radishBOT
Can someone check if this proof is right?
glossy bear
glossy bear
Ok thx
flat frigateBOT
Waes (Wires)
cedar widget
Also $\in$
flat frigateBOT
ExpertEsquieESQUIE
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midnight sonnet
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midnight sonnet
is this a PHP question
cloud hound
What is PHP?
midnight sonnet
What is PHP?
pigeon hole principle
midnight sonnet
idk what the buckets would be tho
quasi bison
maybe you could try to contrive a scenario where NOBODY is surrounded by two boys
glass wasp
I suppose yes, since the only way to avoid that would be to have girl-girl-boy-boy-girl-girl-...
And eventually the last spot would cause 2 boy neighbours
midnight sonnet
maybe you could try to contrive a scenario where NOBODY is surrounded by two boy...
how to do that
will that invlove PHP
glass wasp
how to do that
Like this
glass wasp
I suppose yes, since the only way to avoid that would be to have girl-girl-boy-b...
This
quasi bison
seats 23 and 2 will have to be girls bc then seats 24 and 1 can't have two boy neighbors
midnight sonnet
I suppose yes, since the only way to avoid that would be to have girl-girl-boy-b...
what are the buckets/pigeonholes here?
quasi bison
try to continue filling it in like this
glass wasp
seats 23 and 2 will have to be girls bc then seats 24 and 1 can't have two boy n...
More-or-less the same I suppose?
midnight sonnet
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vague glen
Can someone explain if in the last step, the solution, could it be written as -3sinx³ instead of -3x² as the constant is put in the front , like with 10 here.
I do the formula differently(2nd pic) so naturally the values are placed differently compared to the 1st pic.
I'm stuck pls help
safe radishBOT
Can someone explain if in the last step, the solution, could it be written as -3...
plucky elk
,rotate
flat frigateBOT
vague glen
How'd u get this exact pic to rotate, does it do it automatically
iron estuary
Can someone explain if in the last step, the solution, could it be written as -3...
no it can't be written like that
vague glen
no it can't be written like that
Why not?
iron estuary
the 3 is a constant and can be moved but x^2 is not
it arises from the chain rule while differentiating x^3
so writing -3sinx^3 would be wrong
vague glen
the 3 is a constant and can be moved but x^2 is not
What about 10?
iron estuary
because 10 is just a number, so it’s a constant
you can move constants around not variable dependent factors
vague glen
because 10 is just a number, so it’s a constant
3 is too, I don't understand
iron estuary
yes 3 is a constant but x^2 is not
vague glen
yes 3 is a constant but x^2 is not
Yeah but to obtain 10, 5 & 2 were multiplied which have x⁴ & sinx⁵ as coefficients
Which aren't constant
iron estuary
10 comes from constant × constant (2 x 5) right?
vague glen
10 comes from constant × constant (2 x 5) right?
Hmm, i see
iron estuary
3x^2 comes from constant × variable so only the constant part can be moved around
vague glen
3x^2 comes from constant × variable so only the constant part can be moved aroun...
It can be moved around? 3? How
iron estuary
“moved” just means rearranged in the product
vague glen
“moved” just means rearranged in the product
As in how in this problem?
iron estuary
$-3x^2\sin^2(x^5)\sin(x^3)\quad \text{or}\quad -3x^2\sin(x^3)\sin^2(x^5)$
flat frigateBOT
anflo
iron estuary
or $-,\sin^2(x^5)\sin(x^3),3x^2 ;\text{or}; -3\sin(x^3)\sin^2(x^5),x^2$
flat frigateBOT
anflo
iron estuary
is the fourth one what you mean?
vague glen
or $-\,\sin^2(x^5)\sin(x^3)\,3x^2 \;\text{or}\; -3\sin(x^3)\sin^2(x^5)\,x^2$
WHAT, i asked if it can be written like that -3sinx³
You said no ..
vague glen
is the fourth one what you mean?
Yea
iron estuary
i thought you were removing the x^2 my bad
iron estuary
as long as you don't do that its good
vague glen
So constant can go anywhere regardless
iron estuary
yeah
vague glen
Hmmm
iron estuary
also is this incomplete?
iron estuary
ah okay
vague glen
How're you so good ar maths
It's hard to get answers to such minor questions w/o a teacher so this server is yhr last resort sadly
iron estuary
tbh asking these kinds of questions is how it starts
the rest is just practice and time
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radiant ice
I have two answers: 123° and 66°.
For 123°, if we look at the top left corner, the lower angle is 90° - y, and the upper angle is 90° - x. In a rectangle, and then 33° is the middle angle. These angles form a right angle according to rectangles. So, 90° - y + 90° - x + 33° equals 90°. Adding x and y from the left, subtracting 90° from the right, we get 90° + 33° equals x + y, which is 123.
For 66°, we use the alternate interior angles, which means that y and x are both 33°. Adding both of them gives us 66°.
proud tree
I have two answers: 123° and 66°.
For 123°, if we look at the top left corner,...
123 is right
radiant ice
123 is right
okay thanks
proud tree
alt int angles requires parallel lines
radiant ice
alt int angles requires parallel lines
in rectangles each opposite side is parallel no?
oooh wait
i just noticed
it would be 33 + 90-x
proud tree
for alt int you require red and yellow segments to be parallel
ignore how horribly I've shaded it
radiant ice
for alt int you require red and yellow segments to be parallel
i would still get the same answer if i did it correctly 😂
proud tree
i would still get the same answer if i did it correctly 😂
wym
proud tree
👍
proud tree
I have two answers: 123° and 66°.
For 123°, if we look at the top left corner,...
your reasoning for 123 here is correct
radiant ice
horrible to draw on phone
proud tree
that's correct
proud tree
happens
quasi timber
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midnight sonnet
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midnight sonnet
How to disprove isomorphism between these two
I have checked 11 invariants so far, number of vertices, edges, degree sequence, existence of circuit, euler circuit, hamiltonian circuit, and bipartiteness
They satisfied all of them
cloud hound
Better start comparing adjacency matrices and showing that no rearrangement of rows can make them equal
The way things keep going with you and these invariants 😂
quasi bison
I have checked 11 invariants so far, number of vertices, edges, degree sequence,...
did you compare inner/outer degseq as well
midnight sonnet
did you compare inner/outer degseq as well
well no, lemme try that
does the in/out sequence have to be exactly equal?
quasi bison
obviously yes it does
midnight sonnet
obviously yes it does
one graph has 3 vertices with 2 in and 1 out degree, the other has 4 such vertices
quasi bison
well there you have it
midnight sonnet
well there you have it
but these two graphs have the same seq
quasi bison
redraw the first one by swapping the positions of u1 and u3 on the page
midnight sonnet
although the first graph has a longest path of length 3 while the second graph has the longest path of length 2, so they are not isomorphic
quasi bison
yes or that
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midnight sonnet
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solar hazel
imagine coloring all the edges in K_10 red or blue
safe radishBOT
@midnight sonnet Has your question been resolved?
serene fog
By examining the chest X-ray, probability that T.B is detected when a person is actually suffering is 0.99. the probability that the doctor diagnoses incorrectly that a person has T.B. on the basis of X-ray is 0.001. in a certain city 1 in 100 persons suffers from T.B. A person is selected at random is diagnosed to have T.B. What is the chance that he actually has T.B.?
can someone draw probability tree for this and show pls
safe radishBOT
By examining the chest X-ray, probability that T.B is detected when a person is ...
serene fog
idh my mobile so i cant show my working
nimble vine
By examining the chest X-ray, probability that T.B is detected when a person is ...
no
safe radishBOT
@serene fog Has your question been resolved?
umbral parcel
safe radishBOT
lone void
what have u tried
umbral parcel
i drew a diagram
lone void
send
lone void
huh
nimble vine
Fijix
lone void
can u use different variable for the 2 surfaces
both ur frictions are F(r)
and also how can u say that F(r) = u1 R?
this is only limiting case
iron estuary
fix the notation
lone void
because Fr = Fmax = uR no?
ohk then its correct sorry i didnt read limiting eqm
umbral parcel
ohk then its correct sorry i didnt read limiting eqm
dw dw
umbral parcel
fix the notation
i'll redraw the diagram 😭
iron estuary
the diagram's fine
umbral parcel
is it the Frs?
iron estuary
yeah its the F_r labels
lone void
i dont think u need any more equations..u just have to substitute stuff from here
lone void
FrA and FrB? 😭
u can use small f
iron estuary
FrA and FrB? 😭
yeah that works or just F_a or F_b is good too
umbral parcel
so my 1/2 mg cos theta = Rb sin theta + Fr Cos theta equation is right?
lone void
yeah
umbral parcel
so then FrB cos theta turns into u2R cos theta?
iron estuary
so my 1/2 mg cos theta = Rb sin theta + Fr Cos theta equation is right?
mathematically, its right but just a heads up make sure you're visualizing the perpendicular distances for those moments it's super easy to accidentally swap sin and cos if you treat them like force components instead of distances
iron estuary
lol yeah it can be brutal sometimes
once the moments are written with the correct lever arms try eliminating R_a and R_b
if the forces disappear and only θ is left, you’re on the right track
umbral parcel
if the forces disappear and only θ is left, you’re on the right track
i'm stuck again 😭
iron estuary
oh, what's wrong?
umbral parcel
wait maybe i've got it 😭
i'll come back if i get it wrong
Did I make a silly mistake?
@iron estuary
iron estuary
oh you’re really close just a small slip
umbral parcel
nooo 😭
iron estuary
in the force balance $R_A \neq R_B$
flat frigateBOT
anflo
iron estuary
when eliminating $mg$ use $R_A = mg - \mu_2 R_B$ not $R_B$
flat frigateBOT
anflo
iron estuary
fixing that gives the missing $\mu_1 \mu_2$ term
flat frigateBOT
anflo
iron estuary
yes yes you got it
umbral parcel
where the hell is u1 😭
iron estuary
replace R_a using vertical equilibrium, then simplify
iron estuary
where the hell is u1 😭
it'll come dww don't stop
umbral parcel
replace R_a using vertical equilibrium, then simplify
what does this mean? have I been using horizontal eq?
iron estuary
horizontal equilibrium is just $R_B = \mu_1 R_A$ right?
flat frigateBOT
anflo
iron estuary
vertical is $R_A = mg - \mu_2 R_B$
flat frigateBOT
anflo
umbral parcel
i've made this harder by not including Ra and Rb haven't i?
iron estuary
well i wouldn't say harder... you just took a slightly longer scenic route
umbral parcel
man i don't wanna do this anymore, i've got 5 more of these 😭
iron estuary
oh no where are you stuck
umbral parcel
I’ve gotten here idk if that Rbs and Ras are correct but idk how to get rid of that Ra at the beginning
iron estuary
this is just substitution whiplash
you’re allowed to use mg temporarily if it helps eliminate R_a
think of it as changing variables not undoing work
umbral parcel
ok then (1/2 (mg) - u2Rb)/u1Rb?
iron estuary
yep that idea just keep mg u2Rb and together in brackets 😭
after that i promise it simplifies cleanly
use your horizontal balance
umbral parcel
Am I close? 😭🙏 @iron estuary
iron estuary
$mg = R_A + \mu_2 R_B$
flat frigateBOT
anflo
iron estuary
what have you written?
umbral parcel
i wrote mg = u2Ra + Ra...
iron estuary
well its u2Rb right 😭
verbal void
for this you have to equate torque as well right?
umbral parcel
what the hell is torque? 😭
verbal void
i might be wrong then
umbral parcel
this is making me wanna cry 😭
umbral parcel
Did I make a silly mistake?
looking at this, mg = Ra + Frb no? and Frb = u2Ra
@iron estuary
verbal void
this is making me wanna cry 😭
what i was thinking was that u have two variables r1 and r2 so to solve them youd need 2 equations
umbral parcel
what i was thinking was that u have two variables r1 and r2 so to solve them you...
have you seen what i've done so far? apparently it's right and i'm close but i just don't know how to get there
i've thought about vertical equilibrium and horizontal
used Fr = Fmax = uR
and took moments around A
which is against the floor
verbal void
wait im stupid my bad
iron estuary
$F_{rB} = \mu_2 R_B$
flat frigateBOT
anflo
umbral parcel
Am I close? 😭🙏 <@957875862357831680>
this is where i've gotten to most recently
iron estuary
if that's true then i've done the whole question wrong 😭
you tell me this now 😭
umbral parcel
i thought you saw the screenshots 😭
umbral parcel
I’ve gotten here idk if that Rbs and Ras are correct but idk how to get rid of t...
literally says in top right corner 😭
verbal void
if that's true then i've done the whole question wrong 😭
you did everything else correctly looks like to me
iron estuary
OHH
umbral parcel
i'm going crazy, i just wanna go to sleep 😭
do i have to do the whole question again from the start?
iron estuary
im so sorry 🙏
umbral parcel
im so sorry 🙏
nooo it's not your fault at all 😭
iron estuary
well whatever you've done so far is right luckily
umbral parcel
i should be able to do this
verbal void
btw torque is basicaly moment sorry i confused you
umbral parcel
btw torque is basicaly moment sorry i confused you
oh yh i'm using that
verbal void
u1Ra
umbral parcel
but Fra = Rb
verbal void
because friction is coefficient of friction x normal force at that point
umbral parcel
if left = right
iron estuary
horizontal equilibrium gives Rb = Fa = u1Ra but that doesn't mean Fa = u1Rb
umbral parcel
should i just do the whole question again but with the right equations?
iron estuary
aa im so sorry i feel like ive caused all this 😭
umbral parcel
nonono there's two people helping me now and i still don't fully get it 😭
umbral parcel
aa im so sorry i feel like ive caused all this 😭
more practice, it's okayy
iron estuary
if you don't get the idea behind it you should start over
umbral parcel
ok 💔
iron estuary
i'll actually read everything this time omg 😭
umbral parcel
did you not the first time ⁉️ 😭
iron estuary
i didn't see the right side, well i did kinda but i just didn't notice the A and B
umbral parcel
okok 😭
iron estuary
its okay it'll be quick quick
austere goblet
some channels have like ten of them rotating in and out
but usually courtesy is that if a channel already has good progress with a helper we don't step in unless to correct mistakes
iron estuary
really ten?
umbral parcel
some channels have like ten of them rotating in and out
but usually courtesy is ...
well there was a big mistake but it's ok 😭
iron estuary
😭
umbral parcel
anflo's doing a great job
austere goblet
really ten?
yes for long channels or channels where everyone feels the need to step in for whatever reason
iron estuary
anflo's doing a great job
really i appreciate that but it was my fault
umbral parcel
really i appreciate that but it was my fault
shhh, it's ok
nimble vine
yes for long channels or channels where everyone feels the need to step in for w...
this week there was one with 20
the coordinate system guy
iron estuary
oh wow
umbral parcel
the coordinate system guy
ohh Phoenix?
nimble vine
ohh Phoenix?
nah HereC
austere goblet
ok let's get the channel back on track
opaque fern
omg is there another helper here? 😭
Well this channel has had 300+ messages so far and it seems like the problem is still unresolved
umbral parcel
Well this channel has had 300+ messages so far and it seems like the problem is ...
oh dear, we're getting there 😭
iron estuary
just remember friction is μ × normal at the same contact point
umbral parcel
you did most of the work!
stop lying 😭
iron estuary
now you can sleep peacefully
umbral parcel
i'm contemplating if i want to so another question 😭
umbral parcel
now you can sleep peacefully
yh lol
iron estuary
next one will feel way shorter after this
iron estuary
stop lying 😭
nah genuinely
umbral parcel
next one will feel way shorter after this
the next question had three parts 😭
6, 3 then 8 marks
umbral parcel
okayy, can i find you if i need help again? 😭
iron estuary
sure
safe radishBOT
open wedge
turbid salmon
Given a pyramid S.ABCD with base ABCD being a trapezoid with bases AD and BC, let M be the midpoint of SA. Knowing that BM is parallel to the plane (SCD), calculate the ratio AD/BC.
safe radishBOT
Given a pyramid S.ABCD with base ABCD being a trapezoid with bases AD and BC, le...
turbid salmon
I know I have to use Thales theorem but I just can’t prove it
safe radishBOT
@turbid salmon Has your question been resolved?
lost jewel
Given a pyramid S.ABCD with base ABCD being a trapezoid with bases AD and BC, le...
Have you made a diagram?
If so, send it here, else make one and send it here
radiant ice
try making the base lie in the plane z = 0, and make abcd coordinates
safe radishBOT
safe radishBOT
Available help channel!
latent solar
in this problem's solution: https://wiki.randommath.com/aime/2017/part-i/problem-15
safe radishBOT
in this problem's solution: https://wiki.randommath.com/aime/2017/part-i/problem...
radiant ice
in this problem's solution: https://wiki.randommath.com/aime/2017/part-i/problem...
what are you asking
latent solar
oh give me a sec
i don't understand how they derive the A
like capital A
i know what it's helpful, but idk what they did to get A
cloud hound
(Denominator of s)
latent solar
yea but like the coefficnets of cos and sin are different
cloud hound
Draw a triangle
Hypotenuse A
One side 7sqrt(3)
The other 11
By pythagorean thm, A = what they write
latent solar
well yea i know that
cloud hound
Mark the angle adjacent to the 7sqrt(3) side as alpha
Then cos(alpha) = 7sqrt(3)/A and sin(alpha) = 11/A
latent solar
wait i'm dumb
cloud hound
This gives us the rewriting of the fraction
latent solar
.close
trim slate
hello i need help with factoring, i dont understand how im supposed to know which numbers are the right ones to use if that makes sense.
for example i wanna use 7,8 to factor here but ik it will be smaller numbers but im so confused how im supposed to figure them out
safe radishBOT
hello i need help with factoring, i dont understand how im supposed to know whic...
trim slate
hopefully that makes sense
rustic goblet
hello i need help with factoring, i dont understand how im supposed to know whic...
you need to find the largest square that divides 56 
trim slate
how am i supposed to know what the smallest square is..
is that something you just memorize
primal bone
(well, technically the largest square that divides 56)
rustic goblet
whoops, I dunno why I said smallest 
rustic goblet
how am i supposed to know what the smallest square is..
trial and error is pretty effective when you have small numbers
crimson python
hello i need help with factoring, i dont understand how im supposed to know whic...
I am guessing you wanna know how to divide 56 into its prime factors? like how to write 56 as 2x2x2x7?
rustic goblet
for instance, it can't be 7 because 7^2 = 49 doesn't divide 56
so it's gotta be smaller
trim slate
the only thing is i try to do trial and error but its hard when i dont know whats right or wrong
uneven knoll
hello i need help with factoring, i dont understand how im supposed to know whic...
so you want to know whats the largest number that you could take outside the sqrt?
ie the largest square factor?
trim slate
for instance, it can't be 7 because 7^2 = 49 doesn't divide 56
wait why 2?
uneven knoll
in that case write the prime factorization of the number that you have
trim slate
I am guessing you wanna know how to divide 56 into its prime factors? like how t...
YES THIS
rustic goblet
oh, prime factorization is a bit different 
uneven knoll
so you said that 56=7*8
trim slate
right
uneven knoll
now 7 is prime, but 8 isnt
so write 8 as a product of primes too, how does that look like?
trim slate
so that means i need to go lower?
trim slate
so write 8 as a product of primes too, how does that look like?
okay give me a sec i rlly dont know what primes are either😭
uneven knoll
so that means i need to go lower?
yes, you need to have a product of only prime numbers
primal bone
I'd recommend having a look e.g. at https://youtu.be/XBnUWjo3TgM
uneven knoll
okay give me a sec i rlly dont know what primes are either😭
np, a prime number is a number whose divisors are 1 and itself
trim slate
so that will tell me what primes numbers are?
primal bone
-# And also learning what prime numbers are, since that's paramount to what the goal here is
trim slate
np, a prime number is a number whose divisors are 1 and itself
ohh so basically like the lowest?
uneven knoll
so like a natural number p is said to be prime if the only divisors of p are 1 and p
lets have some examples
trim slate
so basically like 2 and 3 because they cant get lower without 1?
uneven knoll
for example the number 13, the only numbers that divide 13 are 1 and 13 right?
so 13 is prime
primal bone
-# Okay, to paraphrase Inigo Montoya, you keep using that word [lowest] but I don't think you're quite clear on what you mean by that
trim slate
for example the number 13, the only numbers that divide 13 are 1 and 13 right?
okok i see
uneven knoll
so 13 is prime
on the other hand take 12 for example, then its divisors are 1,2,3,4,6 and 12
so it isnt prime
trim slate
so basically what i would need to do, if i assumed 7,8 could work, i need to test to see of both are prime numbers, if not i go lower?
primal bone
-# [you've missed 4, but yes]
uneven knoll
-# [you've missed 4, but yes]
i noticed that and knew that you will probably correct me (since i saw you typing right after i sent my message) so i tried to be fast but you were faster
trim slate
wait so since i wanted to try 7,8 since 7 is a prime number but ik that 8 isnt, wouldn’t i just be able to break down 8
but keep 7?
crimson python
yep
uneven knoll
wait so since i wanted to try 7,8 since 7 is a prime number but ik that 8 isnt, ...
yea, now what does 8 break up to?
uneven knoll
right
trim slate
OKOK i get it but
uneven knoll
so 2 is the only prime divisor
and it occurs 3 times
so 2^3. Now to answer your question, find the largest square in the factorization 7*2^3 of 56
trim slate
so like here, how 5 stayed the same, 7 would also stay the same correct?
trim slate
so 2^3. Now to answer your question, find the largest square in the factorizatio...
okok
uneven knoll
so like here, how 5 stayed the same, 7 would also stay the same correct?
exactly
is 7 the only thing that stays the same tho?
trim slate
no the z^7 aswell right?
uneven knoll
you said that 56=7*2^3 and that 7 is a prime so it certainly isnt a square
trim slate
to find the largest square that’s something ill just have to memorize aswell for future questions?
trim slate
you said that 56=7*2^3 and that 7 is a prime so it certainly isnt a square
right
so prime numbers cant be square?
uneven knoll
now what about 2^3? is 2^3 a square? if no then whats the largest square in 2^3?
uneven knoll
so prime numbers cant be square?
right, because if p is a prime number and its a square then it can be written as p=a^2 for some a y definition of square right?
trim slate
now what about 2^3? is 2^3 a square? if no then whats the largest square in 2^3?
8 since it = 8 right?
uneven knoll
right, because if p is a prime number and its a square then it can be written as...
but then a can neither be 1 nor p because 1^2=1 and p^2 is different than p, which would mean that a is a third divisor of p and thats impossible
crimson python
to find the largest square that’s something ill just have to memorize aswell for...
at most you will have to memorize list of prime numbers which are easy to recognise for small numbers problems only arise when you reach big numbers which look prime but arent like 57
uneven knoll
8 since it = 8 right?
i am not sure what this means
primal bone
/genuine - Do you know what a square number is?
uneven knoll
what is the meaning of a square number
trim slate
i am not sure what this means
2^3 prime is 8 cus it 2^3 =8
trim slate
\/genuine - Do you know what a square number _is_?
not really 😭😭
ive looked it up on a couple past questions so ik like a few
uneven knoll
have you seen the term perfect square before?
trim slate
i think so, when i was breaking down past example questions
uneven knoll
a square number is the same as a perfect square, i just used a different name. Anyways, a square number/perfect square is a number that can be written in the form of a^2 for some a
like 25=5^2, 9=3^2, 49=7^2 etc...
trim slate
ohhh i see, but to confirm “a” in this situation is the perfect square right?
so 5 3 7 are the perfect squares?
or id it 25 9 49 that are the perfect squares?
primal bone
like 25=5^2, 9=3^2, 49=7^2 etc...
It'd be dangerous to try and define "square number" using "squaring"
A square number is a number that is the area of a square with unit-lengths
So for example, if you have a square of side-length 3, it has area 9
So 9 is a square number
trim slate
but doesn’t that mean anything could be a square number?
primal bone
Why do you think that?
trim slate
because you say it’s anything that is side length, couldn’t any number be used for a square?
primal bone
Any whole-number side-length, yes
But the square number is the AREA there
Not the length
primal bone
So for example, if you have a square of side-length _3_, it has area **9**
Notice that I never said THREE was a square number here?
trim slate
but if i did area of square as 4 wouldn’t that mean 16 is also a square number?
uneven knoll
It'd be dangerous to try and define "square number" using "squaring"
hmmm you must have a downside for this in mind to say this but i am not sure what makes it harder to define it like this
primal bone
(because you ought to define what "squaring" even does)
uneven knoll
i see, fair point
primal bone
but if i did area of square as 4 wouldn’t that mean 16 is also a square number?
4 is a square number - it's the area of a square with sidelength 2;
similarly 16 is a square number since it's the area of a square with sidelength 4
So yes, both 4 and 16 are square numbers
trim slate
_4_ is a square number - it's the area of a square with sidelength 2;
similarly ...
wait
crimson python
but if i did area of square as 4 wouldn’t that mean 16 is also a square number?
a sqaure number is a x a where a is a natural/counting number so yes a square of a square number would also be a square, like for a=4 a*a=16 so 16 is a aquare number
trim slate
so does that mean 25 is also a square number?
uneven knoll
A square number is a number that is *the area of a square with unit-lengths*
i always avoid interpretting things geometrically thats why i wouldve never thought about defining it in this way
primal bone
yes
trim slate
SO ANYTHING TIMES ITSELF IS A SQUARE NUMBER?
crimson python
SO ANYTHING TIMES ITSELF IS A SQUARE NUMBER?
yes
primal bone
ye
primal bone
i always avoid interpretting things geometrically thats why i wouldve never thou...
(it at least explains why we call this "squaring" in the first place, though you have a point there)
crimson python
that makes it all make sense
a square number is basically that a number which you can get by multiplying a number by itself
trim slate
omgg that makes everything so much easier
uneven knoll
(it at least explains why we call this "squaring" in the first place, though you...
yea thats definitely true and is a good way to explain it for the first time 
trim slate
so square number and perfect square are the same thing correct? not 2 different terms?
crimson python
yep
trim slate
now what about 2^3? is 2^3 a square? if no then whats the largest square in 2^3?
so there is no square in 2^3?
primal bone
If you write it as 2 x 2 x 2, ...
Can you spot where we're multiplying the same number with itself?
primal bone
2 x 2 IS the square number
trim slate
that just confuses me now
trim slate
2 x 2 IS the square number
right but we cant just forget abt the other 2
what am i supposed to do with it
uneven knoll
so just 2x2^2???
right
primal bone
When we write a x a to mean a times a, THIS we can rewrite as a^2
uneven knoll
and 2^2 is the square number
primal bone
right but we cant just forget abt the other 2
oh I see you meant the whole 2x2x2 bit
yh that works
So you've got 7 x 2 x 2^2 - there any more squares?
trim slate
and 2^2 is the square number
but now that i know this how does it factor unto the equation? or did you ask me this to check that 2^3 was correct?
wait why cant i write it as 2^3 x 7
primal bone
(well, technically the *largest* square that divides 56)
-# [@uneven knoll it's in part because this was the goal...]
uneven knoll
thats what you have reached from all of the work done above
crimson python
yep
crimson python
you should look at 2^3 first
trim slate
wait so 2^3 goes lower?
crimson python
2^3=2 x 2 x 2
uneven knoll
-# [<@769420534537912331> it's in part because _this_ was the goal...]
you mean thats why you chose to define a square number in terms of the area of a square?
trim slate
2^3=2 x 2 x 2
doesnt it just end as 2?
crimson python
2^3=2 x 2 x 2
can you break it down in a way that leads to a sqaure number?
trim slate
so at the end i just remove the ^3 and left with 2 correct?
usually i do that at the end
crimson python
nit exactly
crimson python
so at the end i just remove the ^3 and left with 2 correct?
i am guessing you are saying root(2^3)=2?
trim slate
i am guessing you are saying root(2^3)=2?
honestly no, i just noticed from the videos and examples they just would remove the ^2 at the end
i dont actually know the reason behind it
trim slate
right
crimson python
you can see that there is a square number hiding in 8 what would it?
trim slate
its 4?
crimson python
yes!
crimson python
so can you write root(8) as root(4) x root(2)
trim slate
so i can write it as 4^2?
crimson python
no thats 4x4
crimson python
can you explain more what you mean by 2 separate numbers?
trim slate
like this?
crimson python
yes so now what is root(4) ?
trim slate
2 right?
trim slate
2?
crimson python
thats what you get outside the root
trim slate
ohh 4
trim slate
okay now im lost
crimson python
see root(56)=root(4) x root(7) x root(2) you got till here right?
trim slate
right
crimson python
now you know root(4) is just 2 so you write is at 2
but you cant forget about the non square numbers
trim slate
wait now im confused
crimson python
you may not be able to write them outside the root but they still exist
trim slate
like this?
so after this i just do 7 x 2?
crimson python
so root(4)=2
root(56)=2xroot(7)xroot(2)
crimson python
so after this i just do 7 x 2?
yes but you keep them under the root
trim slate
im confused what keep them under the root means
crimson python
its like you keep root(2) as root(2) as it isnt a perfect
you cant write it as 2 without the root
trim slate
im still confused
what do i do with the equation then
so i should or shouldnt write it as 7x 4x 2
crimson python
you can write root(56)=root(7x4x2)
trim slate
but how do i solve it
crimson python
but 4 is a perfect sqaure so you took it out as a normal number that is 2
trim slate
so i got the root part right?
crimson python
now you are left with root(56)=2xroot(2x7)
trim slate
now you are left with root(56)=2xroot(2x7)
what is rot
crimson python
what is rot
mb spelling mistake
trim slate
wait im confused how is 2xroot(2x7) written like
split kayak
If i might add a detail, you know exponents right?
$16 = 2^4 = 2\cdot2\cdot2\cdot2$
crimson python
so like here, how 5 stayed the same, 7 would also stay the same correct?
just like here you wrote 2 x root(5)
flat frigateBOT
trim slate
im just even more confused where is 16 coming from😭
trim slate
i dont know what exponent rules mean but looking at a picture i know what that is
i know that 16 = 2x2x2x2 = 2^4
if thatswhat ur asking
split kayak
like:
$$2^3 \cdot 2^2 = 2^{3+2} = 2^5$$
$$\left(2^3\right)^2 = 2^{3\cdot2} = 2^6$$
flat frigateBOT
split kayak
The second one is important
trim slate
well kinda, i just relearned it today so its familiar but idk anything beyond the basics
split kayak
well, square roots are a sort of exponent too
to be specific, we say that $\sqrt 2 = 2^{\frac 12}$
flat frigateBOT
split kayak
thats why elements that are a perfect square become an integer
trim slate
im getting more n more confused 😭
split kayak
$$4 = 2^2$$
$$\sqrt 4 = \sqrt{2^2} = \left(2^2\right)^{\frac 12}$$
flat frigateBOT
split kayak
and 2 * 1/2 = 1
so you end up with 2^1 = 2
And this not only applies to 2^2, also for 2^4, 2^6, etc...
trim slate
im really trying to understand but im just getting more n more lost😭
split kayak
could you rewrite 81 in terms of its prime factorization?
trim slate
?!?!??
split kayak
81 is a perfect square btw.
trim slate
whats that gotta do with my question man im getting confused cus im straying more n more away from what im trying to figure out😭
split kayak
understanding how square roots work as exponents basically simplifies all of this to a purely symbolical thing
split kayak
Nope, thats perfect squares.
the square root is an operation that undoes the "squaring" of a number
trim slate
so then how does square roors factor into my equation
trim slate
the square root is an operation that undoes the "squaring" of a number
so then square root of 81 is 9
split kayak
yea
mighty mango
-# you're doing amazing char!!!!!!!
split kayak
now, prime factorization is useful, because it allows you to easily find the square root
If its an integer number
similar example to your problem:
the square root of 20.
do you agree that 20 is equal to 2 * 2 * 5?
trim slate
yea
split kayak
then we can rewrite it as 2^2 * 5
trim slate
i know
i just learned this
this is what i was confused on
but i got an understanding of it now im just trying to incorporate it into my problem
wont work like that
split kayak
can you find the factorization of 56? we will then deal with the z^7
split kayak
Well, then you have to consult this almost by force.
crimson python
so then how does square roors factor into my equation
are you confused about what square roots have to do with your problem?
split kayak
!occupied
safe radishBOT
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).
trim slate
are you confused about what square roots have to do with your problem?
im confused where square root is being incorporated in the problem, we figured out factorization of 56 i thought itd 7x4x2
or 7x2^3
crimson python
im confused where square root is being incorporated in the problem, we figured o...
your entire question is about sqaure roots
trim slate
i thought it was about square numbers
crimson python
genuine question you do know that this symbol is the symbol for sqaure root?
trim slate
i needed help factoring the 56 right, which is square numbers
split kayak
for easy visualization: $\sqrt{2^2 \cdot 2 \cdot 7} = \sqrt{2^2}\cdot\sqrt{2\cdot7}$
flat frigateBOT
split kayak
I hope it eventually makes sense why i divided it like that
crimson python
<@&268886789983436800> troll in help channel
trim slate
so why can i not just use 2^3 x 7 if they equal the same thing?
mighty mango
I've already redirected you to #❓how-to-get-help if you're actually genuine about wanting help. eitherway, the mods will probably warn you 
split kayak
so why can i not just use 2^3 x 7 if they equal the same thing?
Remember the fact that the square root undoes squaring?
exotic anvil
<@&268886789983436800> troll in help channel
Thanks
split kayak
means, if you got $\sqrt{2^2}$, thats just equal to $2$
flat frigateBOT
trim slate
right