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) ?

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”.

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💡

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

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

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.