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That's just noticing that for any number, ##x^{3/2} = x^{1 + (1/2)} = x * x^{1/2}## So ##(y - 2)^{3/2}## can be written as ##(y - 2)(y - 2)^{1/2}##. Oh, I see another issue. ##(y - 2)^{3/2}## shouldn't be under the radical sign. You had ##\sqrt{(y - 2)^3}##. That's equal to ##(y - 2)^{3/2}##...
2. Proof by contradiction - any non-zero number divided by itself is 1

Yes, I'd say that the multiplicative identity can be used here. If C = 1, the multiplicative identity, then B x C = B Using the definition of division (y is defined to be z/x with x nonzero, if xy = z), then C = B/B.
3. A The name of this probability density function

I think that just makes it a truncated normal.
4. Applying the root test

Write ##i## as ##e^{i\pi/2}##. Then it's clear that ##\left |i^n \right | = \left | e^{i n \pi/2} \right | = 1## for any real n.
5. A Optimization with integers as results

You've just discovered by combinatorial optimization is considered a "hard" problem. In general searching for an optimum integer solution takes a very long time as you have discovered. If you can accept merely a pretty-good solution, there are heuristics you can apply. One of them is...
6. B Falling Cat - Rotation with Zero Total Angular Momentum

For a similar analysis with squirrels, check out this hilarious video by Mark Rober, an engineer with way too much time on his hands. Link should take you to 15:59 where the discussion begins.
7. Determine the type of correlation for the two variables given in the data

It has nothing to do with the specific data. If you have something that goes from 0 - 1 with 0 being weakest and 1 being strongest, you can chop it any way you like and define any names you like to break that up into ranges from "weakest" to "strongest". You defined categories for |r|: 0 - 0.5...
8. Finding the area under this unusual curve

Nothing wrong with sharing your solution, but if you post in a forum and expect people to respond, it's generally customary to indicate what sort of response you're looking for.
9. Finding the area under this unusual curve

OK, that seems valid. So now you need to perform the integral. What was your question? I'll also note that you could have just observed that ##A_2## is a square of side 2, therefore its area is 4.
10. I Angular momentum of a disc

Are you asking for an expression for the angular momentum of a disk where the rotation axis is not perpendicular to the disk? Your drawing is not clear. Are ##\omega_x, \omega_y## and ##\omega_z## the cartesian coordinates of ##\vec\omega##? How are ##I_x##, ##I_y## and ##I_z## defined? Can...
11. Problem calculating eigenvalues and eigenvectors

I'm struggling to understand your problem. What are the "grid positions" you're referring to? I believe EIG does not return the eigenvalues / eigenvectors in any particular order. You may be comparing to a plot where they are sorted in order of descending eigenvalue magnitude. Perform that...
12. I Phase Shift of a Sine Wave

You want a different form of the sine wave, sin(k(d - d0)) A positive value of d0 will shift the sine wave left by d0 days, a negative value will shift it right. I'm not sure which of those cell references corresponds to the day number d. But basically you want the shift to be grouped with the...
13. B Will there be a time when drawings stop being useful in mathematics?

As a counterexample, when I took Real Analysis, I found drawings to usually be the key to working out a proof. Even though they were proofs of fairly abstract things generalized to n-dimensional spaces, a little doodling in 2-D was often enough to show me why the theorem was true. In the case of...
14. Simplify An Expression Containing Absolute Values

There's no reason for a "because". (1, 2) is notation for the set of points such that 1 < x < 2. You don't have to do any deduction. This is called interval notation. Square brackets are commonly used for intervals that include the endpoints. If I say x is in [1, 2], that means x lies in the...
15. Do scientists have a social life?

I think you may be doing something like this: - many physicists are introverts - therefore physics causes introversion - or in order to succeed in physics, I must become an introvert Those aren't logical deductions. I personally am extremely introverted. That doesn't mean I hate people or even...
16. CDF of a function of 2 random variables

This is a special case of what is called "order statistics", the lowest, 2nd lowest, 3rd lowest, of a set of random variables. You can google on that keyword to learn more about how they're calculated. ##T \leq x## means either T1 is the smallest and is ##\leq x##, or T2 is the smallest and is...
17. Statistics: Verifying a Probability Proof

It probably wouldn't qualify as a proof, but it would help show you why the probabilities are equivalent. Adding an algebraic argument in terms of a, b and c as in the proof you provided of Theorem 1, would count as a proof.
18. Internal reflection equation question

It's because of the way the angles are defined. Angle of incidence and angle of refraction are the angles relative to the surface normal. That picture defines ##\theta## as relative to the horizontal, which is tangent to the surface. The angle of incidence is ##90^\circ - \theta##.
19. I The domain of the Fourier transform

There is such a thing as a two-dimensional Fourier transform, for instance the spatial Fourier transform of an image. It consists of Fourier transforms applied independently in the x and y directions. Perhaps that's what you need to do here, transform Re(z) and Im(z) to Re(k) and Im(k).
20. How do you find a, b, and show that that 𝑓(𝑥) has a local min value?

Is the attached picture your solution? You didn't tell us the question so it takes some guesswork to figure out what the question was. The attached text says "Since there is a horizontal tangent at the point (1,3)". Was that part of the question, some information you were given? And I have to...
21. B Impossible to find the quantile of any equation

Previous answer deleted. I was confused by your terminology. ##f(x)## is a density, provided you include the restriction that ##x \geq 0##, and ##F(x)## is the cumulative distribution, which is ##\int_0^x f(t) dt##. Sometimes the term "distribution" is used for the cumulative distribution...
22. Torque on an atom due to two infinite lines of charge

I think you will find that there are relevant equations in your textbook. You're asked to find a force. On what due to what? What equations are relevant to the force between those sorts of things?
23. Physics problems with Newton's law of gravitation

This picture is incorrect, as you can see if you study it closely. You've drawn a ray which is the hypotenuse of a right triangle. But notice that in order to reach the far vertex of that particular triangle, your ray has to penetrate the surface of the planet. That is not the ray that defines...
24. B What is the meaning of 'the speed of light is constant'?

Yes. No. Suppose I say an object traveled 10 m in 1 second. I say the speed is 10 m/s. You say I'm wrong, according to your measurement the object only traveled 9 m in my experiment. And it took only 0.9 seconds to do so. You say the speed is 9/0.9 = 10 m/s. We disagree on the distance and...
25. Implicit Differentiation

Wait. Are you differentiating with respect to temperature (T) or time (t)? Don't keep switching back and forth between them, choose one symbol. it's very confusing. Adjusting the formatting to what I think that says based on the brackets, I think that line says this: \frac{nR - \frac {P +...
26. B A doubt on Rutherford's Scattering experiment

Your bullet analogy is imperfect but can work. Post the text and your question HERE if you want to know more.
27. I Explanation of Galilean transformations

Keep things simple. You have a pair of x and y axes. You have another pair of x and y axes. One of them is moving off to the right. What you're doing in these transforms is asking what coordinates are assigned to any given event by two different observers. Let's say B (the primed frame) is the...
28. Python Can I ask some basic Python installation questions?

OK, so it sounds like Anaconda has a lot of popular packages right out of the box. Does it also make it easy to add additional packages? Also, what do people like for 2D and 3D graphics? Matlab is still my favorite for both. It never looks that great with the default settings but if you tweak...
29. How did Galileo manufacture his telescope?

Did Galileo manufacture his own telescopes? I thought I read there was a lens maker who had invented the telescope and Galileo purchased his lenses from that guy. My brother is an amateur astronomer and I remember when we were teenagers he had a glass blank mounted on an oil drum, that was...
30. How Would Earth Handle Alien Money?

More than one sci-fi author has tackled the question of what aliens would trade for whatever they want from Earth. Two that come to mind are "The Big Front Yard" by Clifford Simak and "Craphound" by Cory Doctorow. In Craphound the aliens are bringing big technological ideas that make people...