In typesetting the following in a,  \sqrt[n]{a \cdot b} = \sqrt[n]{a} \cdot \sqrt[n]{b}   … the second radical is being typeset higher than the first:    Why? And how can I fix it?     MWE:  \documentclass[12pt]{article}  \begin{document}  \begin{displaymath} \sqrt[n]{a \cdot b} = \sqrt[n]{a} \cdot \sqrt[n]{b} \end{displaymath}  \end{document}
This part C\frac{1}{\sqrt{G}}\sqrt{\rho} doesn't seem to be on the same height as this part \sqrt{b}\sqrt{l} when compiled. What am I doing wrong?  \begin{flalign*} T=C\frac{1}{\sqrt{G}}\sqrt{\rho}\sqrt{b}\sqrt{l} &amp;&amp; \end{flalign*}
Need to proof by induction (calculus1 n and n+1) Are software titles italicized? The titles of freestanding works (books, movies, plays, albums, etc.) are italicized; smaller parts within these works (chapters in a book, articles in a magazine, songs within an album) are not (they are put in quotes, instead).  How are software titles treated?  My understanding is that if a particular program resembles a book or a dramatic work (like a video game), its title is italicized -- otherwise, it is treated as a proper noun, just like any other product.     I played Grand Theft Auto, instead of reading "Democracy in Ancient Greece" in Microsoft Encarta and preparing my presentation in PowerPoint.   Is this approach correct? Is it possible to test apex that relies on field history tracking? Running 64 bit programs on a 32 bit system Open source computer algebra systems for general relativity I would like to use an open source computer algebra system (CAS) for research in general relativity. Kindly suggest a good choice between Ipython and Sage. I am more interested in the theoretical aspect rather than numerical computations. I have the following code and I expect the resulting angle to be 135 degrees but whatevet I try, I get 90.  \documentclass{article} \usepackage{tikz} \usetikzlibrary{calc} \begin{document} \begin{tikzpicture}  \coordinate (A) at (1,0); \coordinate (B) at (-1,1);  \pgfmathanglebetweenpoints{A}{B} \edef\angleAB{\pgfmathresult} \node at (0,0) {\angleAB};  \end{tikzpicture} \end{document}   I also tried the following command without knowing what it means, it also prints 90 except one case. The only working case is when A=(1,0) and B=(-1,0), which it prints 180.  \pgfmathanglebetweenpoints{\pgfpointanchor{A}{center}}{\pgfpointanchor{B}{center}}   Why is that? Couldn't find any reference document for "pgfmathanglebetweenpoints" by the way, that's why I am asking here.  EDIT: B is corrected from (-1,-1) to (-1,1) When I watch the news I've noticed that the Gallup polls for things like presidential elections have [I assume random] sample sizes of well over 1,000.  From what I remember from college statistics was that a sample size of 30 was a "significantly large" sample.  It was made to seem that a sample size over 30 is pointless due to diminishing returns. I needed to create a list of lists in Python, so I typed the following: my_list = [[1] * 4] * 3  The list looked like this: [[1, 1, 1, 1], [1, 1, 1, 1], [1, 1, 1, 1]]    Then I changed one of the innermost values: my_list[0][0] = 5  Now my list looks like this: [[5, 1, 1, 1], [5, 1, 1, 1], [5, 1, 1, 1]]    which is not what I wanted or expected. Can someone please explain what's going on, and how to get around it? I'm trying to understand an algorithm that tries to map points from a lie group to its lie algebra using the exponential map. The background is the representation of 3d coordinate transformations as a lie group.  I can imagine, how one can map points on a manifold (eg a sphere) to the tangent space (tangent plane) at some point x on the sphere. I can imagine to create the tangent space by computing derivatives at that point x. But I cannot imagine how the exponential of some matrix is giving me that tangent space?  How can I envision the exponential of a matrix? Is there some picture or analogy I can use as intuition? I'm taking an electronic circuit analysis class, and I was asked to give the color of the 3rd band of a 1MΩ resistor. My answer was blue, thinking it could be black-brown-blue (01 * 1MΩ), but the automated quiz said the correct answer was green (brown-black-green).  So I did some research, thinking there were just multiple correct answers, and I read in a few places that the first band on a resistor is never black (0). Why is this? Does a black first band have some other meaning? It would really help me remember it if I knew the history or reasoning behind it.  This question has been answered. For further reading on zero-ohm resistors mentioned in the answers and comments here, I found these questions and answers helpful: When is a clause not essential? I am having trouble figuring out when to use commas to set off &quot;nonessential&quot; information.  Sometimes it's obvious:  Bob, who is thirty years old, is an alcoholic.  But other times I'm not sure:  The day he quits drinking * he will start a llama farm He has his heart set on owning El Duderino ranch * in New Mexico.  In the first case, my ear says that there should be a comma at *, even though information before it seems essential to me.  In the second case, the stuff after * is not essential, and yet it seems a little much to use a comma. Even that last sentence I wrote confuses me.  &quot;In the second case&quot; seems essential but I used a comma.  Is this correct? Does it depend on personal style and the length of the clause?  Or perhaps I'm misinterpreting the meaning of &quot;essential&quot; in this context? I'm completely new and lost in Ubuntu 12.04, I'm having trouble navigating to my Sublime folder which I know is in my /opt folder. Yet I've tried to no avail to navigate into it via Terminal window. The directory is colored in blue, no idea what this means...but I am able to access the folder using the GUI explorer...?  k@k-Ubuntu:~$ cd /opt k@k-Ubuntu:/opt$ ls Sublime Text 2 k@k-Ubuntu:/opt$ cd Sublime Text 2 bash: cd: Sublime: No such file or directory k@k-Ubuntu:/opt$ cd /Sublime Text 2 bash: cd: /Sublime: No such file or directory I'm reading "BASH pocket guide of Oreilly". It said:      The process ID of the current Bash process.   In some cases, this can differ from $$.   Above explanation , explained $BASHPID variable.  Question: which cases? Adding the PHP json extension Render turns out plain white For some reason, when I render out the image, it is plain white. I've checked the sky settings, and I tried applying textures, but nothing seems to change.  How can I fix this? I  here on MSO today, but it was found to have been asked .  I'm not concerned by this (better to be found by more than one user) but, I noticed that the wording in the closure statement is factually incorrect:    No it doesn't have an answer - Not even a sniff of one!  This statement, logically, should be factually accurate and reflect the existence of answers and include singular/plural references to them - or at a minimum use language that covers both scenarios.  Edit: As a result of my question,  and the bug in question has been remedied, but this question still remains valid. Can't access pre populated SQLite database using PhoneGap/Cordova in Android Visiting USA as a German with Iranian parents Biblatex ieee incorrect display of volume and issue number When in Journey Builder what is the difference between journey data and contact data? I am creating a JB with an engagement split. What is the difference between Contact Data and Journey Data. ie what will I see listed in the two groupings. If I choose the journey data will this only be as up to date as the time that the data entered the Journey?