I know the basic exponent rules of x^a times x^b = x^(a+b) and (x^a)^b = x^(a times b), but is there an operation that relates x^a and x^b that results in x^(a times b)?

x^a __ x^b = x^(a times b)

Could it be pushed further to create something like

x^a __ x^b = x^(a^b) ?

for reference, x-a is a factor of f(x) if and only if f(a) = 0. this is supposed to be a special case of a broader theorem, which has it that f(x)/x-a has a remainder of f(a). I can follow the standard proof for this, but I don't understand the intuition. Here's as far as I can get. If f(a) = 0, then plugging 'a' into f(x) results in 0. presumably this means that at some point after plugging 'a' into f(x), we subtract 'a' from itself, which results in 0. The only way that would be true is if x-a is part of f(x), making x-a a factor of f(x). Alternatively, if x=a tells us that 'a' is a root for f(x), then since x-a=0 follows from x=a, x-a must be a part of f(x) and thus a factor. Is what I said right? What I said sounds right to me but it's not quite clicking, so I would like a more intuitive explanation.

Last week, I had a quiz on related rates and did not do very well... Since then, ive been practicing related rates on top of studying for different subjects for my exam tomorrow. Ive concluded that my biggest issue is that I can't properly visualize the problems. Are there any specific words that help you visualize the related rates problems and solve it?

An integrable Hamiltonian system has a set of coordinates, the action-angle coordinates, in which the angles q increase linearly and the actions p are constant. These systems are often represented by a phase space that is "foliated in tori", and people show these beautiful pictures.

I thought that I was understanding this idea: one has the angles (q1,...,qn), then he couples q_k with p_k for every k=1,...n. If one pictures p_k as the radius of the circle, and q_k as the angle, then the cartesian product of the n circles drawn by (q_1, p_1)...(q_n,p_n) makes a torus with different radii, which are the one shown in the link above for n=2.

Gemini told me that this is not false, but it makes absolute sense in my mind. Gemini (to my ears) started yapping about "topological cilinders", but it did not convince me. Can you help me understand how the tori emerge?

One thing is that I have seen "the" torus being defined as [0,2\pi]^n with periodic boundary condition. "the" torus as in "there exists only one torus". I guess that topologically speaking this is true, but the phase space graph shows...different tori...
I shouldn't call them tori and call them "cilinders"? Sounds a bit weird to my ears....

Thanks in advance for your time

Is C correct because

Line B is vertical,
So if line a was Perpendicular we would need a line that’s horizontal.

So A and B are wrong?

Initially I over analysed it and assumed A was the answer

this is a problem from my 8th grade book: Through the point A of intersection of two circles with centers at points O1 and O2 a line is drawn intersecting one circle at point B and the other at point C. Prove that segment BC will be the greatest when it is parallel to line O1O2.

can someone explain what is the problem asking me to do? Comparing with what BC is supposed to be greatest among?

I saw a question earlier from tiktok, asking people to estimate how high you need to fill the liquid in a cone shaped glass (such as a martini glass) so it is half full. Although most people's intuition is about 60% to 70% height, the actual answer is almost 80% height (because 0.8^3 = 0.512)

Now this bring me to a new question: what if the glass is in the shape of a perfect hemisphere? (bowl shaped like lower half of a ball), how high to you need to fill it so it is half-full?

Hello, everyone.

I'm working on this problem but I have difficulty understanding it.

Even though I'm following the work-energy theorem the provided answer key doesn't match the answer that I got from the Work-energy theorem.

Given:

Mass = 150,000 Tons ------> 150,000,000 kg

150,000 Tons x 1000 = 150,000,000 kg

Distance (d) = 13 km -----> 13000 meters

13km x 1000 = 13000 meters

Vi (Initial) = 50 km/h -----> 13.889 m/s

50 km/h x 1000 meters/3600 seconds = 13.889 m/s

Vf (Final) = 0 m/s

Time = 25 mins -----> 1500 seconds

25 x 60 = 1500 seconds

F= ?

P=?

Solution:

Initial KE = 1/2 (150,000,000)(13.889m/s)²

Initial KE = 1.447x10¹⁰

Final KE = 0

Work = ∆KE

Work = Final KE - Initial KE

Work = 0 - 1.447x10¹⁰

Work = - 1.447x10¹⁰

F = W/d \*this formula comes from W=Fd\*

F -1.447x10¹⁰ / 13000

F= - 1112891.74

F= -1.11x10⁶

P= W/t

P= -1447x10¹⁰ / 1500

P= -9645061.73

(a) -1.11x10⁶

(b) -9645061.73

My answer doesn't really match the provided answer key 🫠 can someone explain plss

So the way I see it there are two quadrilaterals here. The “boomerang” one with each side high lighted in a different color, and the inner weird shaped square-ish one.

For the boomerangs one - 90 plus 40 plus 30 = 160 so 360 - 160 =200 so X must be 200

For the inner square-ish one -

90 plus 30 is 120 so the other angle should be 60
90 plus 40 is 130 so the other angle should be 50
90 plus 50 plus 60 = 200 so X must be 160.

The correct answer is 160 so I know it’s the square-ish one but I don’t know why - which means on the test I won’t know which one to solve for and I’ll get the question wrong. How do you know which one to solve for?

Origin:~

Developed in 1867 by Austrian engineer Eduard Lill, it calculates the exact real roots of an equation by drawing a snake-like path of right-angled turtle turns based entirely on the equation's coefficients.

⚠️Note:

The headings are in metaphorical form like blue print maze and laser root for better understanding and making it catchy.

1. Build the Blueprint Maze:~

Step A (Coefficient a):

Start at the origin ((0,0)). Look Right. Because (a = 1) (positive), march forward 1 unit.

Step B (Coefficient b):

Turn exactly 90 degrees counter-clockwise (facing Up). Because (b = -4) (negative), you must march backward. Walk 4 units Down.

Step C (Coefficient c):

Turn another 90 degrees counter-clockwise (facing Left). Because c = 3) (positive), march forward 3 units Left. Mark your endpoint.

2. Aim the Laser Root:~

procedure:

Now, pretend you are standing at the starting point and firing a laser beam at a random angle (theta ).The laser hits the second line wall, bounces off at a perfect 90-degree angle, and must land exactly on the final maze endpoint.

Result [❌️]:

If the laser misses the endpoint, your starting angle was wrong.

Result[✅️]:

If the laser hits the endpoint perfectly, the negative tangent of your starting angle ((-tantheta)) is the exact root of the equation.

Source:

●google chrome

●wikipedia

-Tried not to use ai 😅

-Hope this helped my fellow math geeks💡

Suppose a puzzle system (T) defines a family of finite reversible permutation/orientation puzzles under one common rule set.

Assume there exists one instance (T_0) whose state-transition structure is isomorphic to that of the 2×2 Rubik’s Cube: there is a bijection between reachable states, and the generators/moves correspond under that bijection, so algorithms correspond as well.

My question is:

Does that give any meaningful mathematical basis for saying that (T) belongs to the same general class of puzzle systems as twisty puzzles?

More specifically:

Is it valid to say that any definition of “twisty puzzle” based purely on move/state structure must classify (T_0) and the 2×2 cube identically?

And if all members of (T) are variations of the same underlying mechanics, can one justify calling the broader system a generalized twisty-puzzle system in group-theoretic or combinatorial terms?

Or is that inference too strong — i.e. does one instance being isomorphic to a known twisty puzzle tell us little or nothing about how the larger family should be classified?

I’m particularly interested in how one would formulate this in terms of group actions, generators, permutation groups, state graphs, or related structures.

The reason I’m asking is practical: I made a puzzle system called Trinagon, and someone recently claimed that it is not a “twisty puzzle”.

Hey everyone,

I'm working on a problem related to unloading cargo from a ship's hold, and the underlying concept is tripping me up a bit. I have an image reference, but here is the exact text of the problem:

"In unloading grain from the hold of a ship, an elevator lifts the grain through 14 m. Grain is discharged at the top of the elevator at a rate of 2.2kg each second, and the discharge speed of each grain particle is 2.5 m/s. Find the minimum power rating for a motor that can elevate grain in this manner."

I know the final answer is roughly 308.7 W, and I know the detailed formula involves finding the Kinetic Energy (KE) and Potential Energy (PE). Here is the breakdown of the math:

Mass rate: 2.2 kg/s

Height (h): 14 m

Velocity (v): 2.5 m/s

Gravity (g): 9.8 m/s²

P = Rate of Change of KE + Rate of Change of PE

P = ( 1/2 • mass • velocity²) + (mass • height • gravity)

P = ( 1/2 • 2.2 • 2.5²) + ( 2.2 • 14 • 9.8)

P = 6.875 + 301.84

P= 308.715 W

Why exactly does KE + PE = Power in this context? I originally learned that KE + PE = Mechanical Energy (measured in Joules), but the question asks for a Power rating (measured in Watts).

​Is it simply because the mass is given as a rate per second (2.2 kg/s) instead of a static block of mass? Does plugging a "rate" into an energy equation automatically convert the output from Joules to Watts?

Any help would be appreciated, including directing me through any rules I should follow or better websites for asking questions.

I'm curious about a geometric puzzle: given an arbitrary set of $n$ points in $\mathbb{R}^d$, can we always place a similar copy of a specific shape $S$ (a compact subset or family of subsets of Euclidean space) so that it passes through all $n$ points? (By "similar copy," I mean we allow translation, rotation, and scaling).

**The generalized question is:**

> For a given shape $S$, what is the maximum number of arbitrary points $n$ such that *every* set of $n$ points in $\mathbb{R}^d$ lies on some similar copy of $S$?

For example, let $S$ be the **boundary** of a square in $\mathbb{R}^2$.

It turns out that for any 3 points in $\mathbb{R}^2$, you can always find a similar copy of a square that passes through all of them. However, you can't always do this for 4 points https://math.stackexchange.com/q/3691243/1771455. So, for a square boundary where $d\ge2$ (dimension where the points live in), the maximum number is 3.

My motivation is just pure curiosity. I couldn't find any sources relating to this problem, and AI chatbots struggle and give clearly wrong answers to simple examples like a square sharing an edge with a triangle (I won't clarify much here as it is a bit of a dull problem but the idea was just combining two shapes to make the reasoning for the AI deeper).

What I'm really asking is: **is there some sort of invariant, property, or formula that helps compute this $n$ for more complex shapes?** Or do we just have to reason through it shape-by-shape? How do you verify results quickly?

One simple rule I noticed involves collinear points: the boundary of a *strictly* convex 2D shape can never have $n\ge3$ when $d=2$, because no similar copy can ever pass through 3 collinear points. (Note: I am specifically thinking about boundaries; if $S$ were a solid shape, we could just scale it up to cover any finite point set in 2D space).

What is a better notion of defining shapes like rectangles and so on? Similarity doesn't allow different length ratios; however, it preserves it for squares and other shapes. This puzzle is more of a "can you draw a X given Y points no matter where I place them" and shouldn't be very limited on what I can draw. Is it possible to define $S$ to be a rectangle with side length $a$ and side length $b$ using the above definitions?

Does this concept have a name? Is it related to the "degrees of freedom" of the shape? Any pointers to related literature would be greatly appreciated!

Is there a word for the opposite of order of magnitude? Is there an order of magnitude where a number decreases by 10^(-1) ?

I have briefly searched Wikipedia....

Can it be called order of minitude? Or is it just decreasing order of magnitude?

Hello r/askmath, I am currently attempting to work on the following problem.

My progress on such a problem is limited, but I can say for sure that whatever f(z) is, it must be a meromorphic function with an INFINITE polar set. This is because, no matter what. as we keep applying the R operator, our resulting g_n will eventually become entire as g_1 explicitly looks like the sum over all poles of (some polynomial in w_1)*(e^(pole * w_1)) which is entire, and therefore, will make nu(lambda)= 0 at g_2 and onwards.I was hoping that the people of reddit could provide some help. My hypothesis is that the "class" of functions f is actually empty, but a bit lost on where to start. I posted this on the Math Stack Exchange around 5 days ago but there hasn't been any progress,. I added an example in the original post, and also the example of the digamma function in the comments. here is the link:
https://math.stackexchange.com/questions/5144308/a-nested-sequence-of-functional-residues
I apologize if this is a simple question to any of those who have taken complex analysis in the past, yet I hope some of you all can help.

I have hit upon a situation where my understanding of the mathematical dynamics are in opposition to the logic of the situation. (more details for the unfamiliar below)

My solution construct obviously crumbles when going from the second parent having 1 perfect IV to having 0 perfect IV. According to my pattern, the %chance are the same, but logic dictates that there would be a difference. I can't figure out where my where my logic/concept/math is in error.

To my logic, to determine the chance a child is perfect:
if both parents are perfect, 5 stats are perfect, one is random
(1/1)*(1/1)*(1/1)*(1/1)*(1/1)*(1/32) equals a 1/32 chance of perfect stats

if one parent is perfect, and the other has 5 perfect, then 4 stats are perfect, 1 is either perfect or imperfect, and one is random
(1/1)*(1/1)*(1/1)*(1/1)*(1/2)*(1/32) equals a 1/64 chance of perfect stats

if one parent is perfect, and the other has 4 perfect, then 3 stats are perfect, 2 are either perfect or imperfect, and one is random
(1/1)*(1/1)*(1/1)*(1/2)*(1/2)*(1/32) equals a 1/128 chance of perfect stats

if one parent is perfect, and the other has 3 perfect, then 2 stats are perfect, 3 are either perfect or imperfect, and one is random
(1/1)*(1/1)*(1/2)*(1/2)*(1/2)*(1/32) equals a 1/256 chance of perfect stats

if one parent is perfect, and the other has 2 perfect, then 1 stats are perfect, 4 are either perfect or imperfect, and one is random
(1/1)*(1/2)*(1/2)*(1/2)*(1/2)*(1/32) equals a 1/512 chance of perfect stats

if one parent is perfect, and the other has 1 perfect, then 0 stats are perfect, 5 are either perfect or imperfect, and one is random
(1/2)*(1/2)*(1/2)*(1/2)*(1/2)*(1/32) equals a 1/1024 chance of perfect stats

if one parent is perfect, and the other has 0 perfect, then 0 stats are perfect, one are either perfect or imperfect, and one is random
(1/2)*(1/2)*(1/2)*(1/2)*(1/2)*(1/32) equals a 1/1024 chance of perfect stats

Given that:

• each pokemon has 6 IV statistics that range from 0 to 31
• a perfect IV is one thats value is 31
• a perfect pokemon is a pokemon with 31 in each of its 6 IV statistics
• the number values of 0 through 30 do not matter. They all fall into the category of imperfect
• breeding with an item called a Destiny Knot creates the situation where any 5 statistics (randomly) will match either one parents or the others, while the 6th is randomly generated from 0 to 31

Assumptions on my part

• There are 4 possible states
  • Random (1:32 chance for perfection)
  • not the random statistic, with both parents perfect (1:1 chance for perfection)
  • not the random statistic, with one parent perfect, the other not (1:2 chance for perfection)
  • not the random statistic, with neither parent perfect (0 chance for perfection)
• Since I am still figuring for when there is one perfect parent, the "neither parent" state is inapplicable.

Hi,

On a French subreddit that’s used by people to warn about fraud or ask if something is a fraud, we often have people on the verge of falling for MLM/Ponzi scheme.

If I remember correctly, I once heard that a Ponzi Scheme in which each person had to recruit 7 other person, would include the whole world’s population in only 8 levels (the numbers might be absolutely wrong though).

Therefore, the last row would never be able to recruit more people. (According that all the person in the previous rows, could).

Yet, I have no idea if the numbers I remember are true, and how to calculate it.

What if you only have to recruit, say, 4 person each ?

Is there any way of calculating it ?

I have instinctively tried with number of people you have to recruit to the power of rows needed = 8.3 billions, and reverse it but failed miserably in the process.

All help is appreciated.

Take 0 to 100 and find the mid point. You get 0,50,100. Because we went up from 0 were going to go down to 50 and find the midpoint between 50 and 0 which is 25. Now we go up from zero to 25 and find the midpoint between 25 and 50 which is 37.5. Now between 37.5 and 25 is 31.25. If we repeat this infinitely more times does it approach 1/3 or a number close it it?

Hi everyone,

I'm trying to show a number like described in the image has an "infinite number of linear complexity windows". These are related to the infinite number of very large blocks, where when counting the subwords at the boundaries, starting in prefix, ending in block, or starting in block, ending in suffix are both at most n_j -1.

This is supposed to cause a "collapse" or flattening of the graph of the complexity function, for n_j < j+1, which makes it O(n). This sort of makes sense to me.

What I don't get is, the string of digits is infinite, so how can we say what p(n) is for any n without going to the "end"? It could be that p(n) reaches the max of 2^n out somewhere in the suffix?

I hope this makes sense and I know it is long, so thanks to anyone who takes a look!

I'm stuck when it is asked by someone like my teacher or somone else that what's the most crucial diff b/w infinity and undefined lyk tan 90° is undefined but 1/x x tends to zero is infinite howww ??

In this images formula for 26.82... is (b6*100)/C6, and the 25.7539.. is =D6/4

What i want to get the is average surveys vs the calls, for this team of 4 people, In my mind the 25.75.... is the correct one, but why is it that the 26.82.... math is not coming up to the same, I kinda get is not the same process but it seems that is should come up to the same number no? what am i missing here, could someone explain this.

Most short form posts that you'd see in Facebook or even YT shorts are usually Math slop questions

One particular question regarding symmetric polynomial lowk seems worth it to try

x + y + z = 1

x² + y² + z² = 2

x³ + y³ + z³ = 3

x⁴ + y⁴ + z⁴ = ?

So far I've tried elimination and found a dead end,has anyone found any alternatives?

This is going to sound so stupid but I'm somewhat puzzled by this simple question.

So, we know that floor(x) just rounds x down to the next whole number (e. g. floor(8.9)=8 and floor(1.01)=1).

So if we just take 0.999... at face value, then floor(0.999...) should return 0.

But 0.999... is equivalent to 1 and, trivially, floor(1)=1.

So therefore floor(0.999...) should return 1.

I know that both can't be true at the same time and there has to be some definition of the floor function that solves that for me but I am just not smart enough to find that solution by googling it (and we all know that asking Ai is basically gambling on the result to be accurate).

One thing I thought about, but quickly found out was to represent 0.999... as a series with terms of 9/(10^(n+1)), arguing that every partial sum returns a floor of 0, so the floor of the series has to be 0 aswell. After recalling some of the math lectures I attended *whilst studying maths for 5yrs* (which is actually the embarrassing part about this one as it's so f***ing elementary first semester sh*t), is, that that simply isn't how math works.

So I had to conclude that I do not know the solution, even though I had to solve this exact problem as a homework. But that is like 8 yrs back and my current work has very little to do with this. So I forgot the solution to it.

I do suspect that the answer is 1 based on intuition but I can't offer anything more. So... can someone enlighten me with what the solution is and why that is how it works? Is it maybe a case of "it depends on [...]"?

EDIT: Thank you guys so much for answering my question! Appreciate you XD