Do exponent rules follow different rules from radicals
Why $\sqrt{-1 \cdot {-1}} \neq \sqrt{-1}^2$?
Last digits, numbers Can anyone please help me?  1) Find the last digit of $7^{12345}$   2) Find the last 2 digits of $3^{3^{2014}}$.  Attempt: 1)  By just setting the powers of $7$ we have $7^1 = 7$, $7^2=49$, $7^3=343$, $7^4 = 2401$, $7^5 = 16807$, $7^6 = 117649$, $\dots$  After the power of $4$, the last digits will repeat. Then by noticing the pattern the digits will end in $7,9,3$ and $1$.  Then we can divide the exponent $(12345)$ by $4$ since this is the cycle that makes it repeat. Then $12345 : 4$ has remainder $1$. So $7^1 = 7$ is the unit digit to $7^{12345}$.   So the last digit is $7$.   I know how to do it like this, the problem does not state how to find the last digit, but I know it has something do do with Euler's theorem.  for part $2$) I don't know how to start. Can anyone please help me?   Thank you for the help. Is irrational times rational always irrational? Is an irrational number times a rational number always irrational?  If the rational number is zero, then the result will be rational. So can we conclude that in general, we can't decide, and it depends on the rational number? How to make the function of the planes from the lorenz attractor I want to subtract a constant number from my data to get new data file. For example, I want to subtract  10 from my data.   data.txt  Id  a   b   c   d   e 1   1   2   3   4   5 2   2   3   4   5   6 3   3   4   5   6   7 4   4   5   6   7   8 5   5   6   7   8   9 6   6   7   8   9   10 7   7   8   9   10  11 8   8   9   10  11  12 9   9   10  11  12  13 10  10  11  12  13  14 11  11  12  13  14  15 12  12  13  14  15  16 13  13  14  15  16  17 14  14  15  16  17  18   Required output:  out.txt   Id  a   b   c   d   e 1   -9  -8  -7  -6  -5 2   -8  -7  -6  -5  -4 3   -7  -6  -5  -4  -3 4   -6  -5  -4  -3  -2 5   -5  -4  -3  -2  -1 6   -4  -3  -2  -1  0 7   -3  -2  -1  0   1 8   -2  -1  0   1   2 9   -1  0   1   2   3 10  0   1   2   3   4 11  1   2   3   4   5 12  2   3   4   5   6 13  3   4   5   6   7 14  4   5   6   7   8 How to fix java.lang.UnsupportedClassVersionError: Unsupported major.minor version How to get hardware specs of Android device? How can I find information on the hardware of my Android device like processor architecture, RAM, etc? maximum of two uniform distributions I have a question. Let's suppose that the two random variables $X1$ and $X2$ follow two Uniform distributions that are independent but have different parameters:  $X1 \sim Uniform(l1, u1)$  $X2 \sim Uniform(l2,u2)$  If we define X3 as the maximum of X1, and X2, i.e., $X3 = max(X1, X2)$, what kind of distribution would it be? It is certainly not a uniform and I can calculate the cumulative probability, i.e., $p(X3 &lt;= a)$, because $p(X3 &lt;= a) = p(X1 &lt;= a) p(X2 &lt;= a)$. But, I wonder if there exists any density function that can model X3. I'm very new to classical propositional logic, and my lecturer is using a system with the following axioms:  A1. X→(Y→X)  A2. (X→(Y→Z))→((X→Y)→(X→Z))  A3. (¬Y→¬X)→(X→Y)  Use uniform substitution and Modes Ponens (MP) as rule of inference. Deduction Theorem also allowed.  I'd like to deduce the following:  (a) X, Y ⊢ ¬(X→(¬Y)) (conjunction, but expressed in terms of NOT and IMPLIES)  (b) X→((X→Y)→Y)  (c) (X→Y)→(((¬Y)→X)→Y)  (d) (((¬Y)→X)→Y)→(X→Y)  Remark: The last two are to show that the axiom (A3) can be replaced with a different axiom:  B3. (¬Y→¬X)→(((¬Y)→X)→Y).  Solution: Bram28 proves (b) and (c) below. I demonstrate (a) and (d).  (d) is alright; first, I claim that X, ((¬Y→X)→Y) ⊢ Y.  Axiom (A1) gives ⊢X→(¬Y→X) so that ((¬Y→X)→Y)⊢X→Y by Hypothetical Syllogism (HS), which Bram28 shows to be valid.  Now that we have ((¬Y→X)→Y)⊢X→Y just apply the deduction theorem to get what we want: ⊢((¬Y→X)→Y)→(X→Y).  (a) is slightly trickier. Axiom (A1) gives ⊢ Y→(X→Y) so that X, Y ⊢ (X→Y).  Next, by part (b), we have as a theorem that ⊢X→((X→¬Y)→¬Y).  Therefore X, Y ⊢(X→¬Y)→¬Y.  But ⊢(¬¬(X→¬Y))→(X→¬Y) by DN Elim, which Bram28 shows to be valid.  By HS, X, Y ⊢(¬¬(X→¬Y))→¬Y.  By Axiom (A3) and MP, X, Y ⊢(Y→¬(X→¬Y)).  Also, rather trivially X, Y ⊢Y.  Finally by MP we conclude X, Y ⊢¬(X→¬Y) as required. Viewer-independent copyable spaces at the beginning of a line? I need to include a couple of Python code listings, where the indentation of the lines (using some number of spaces) is significant. I would like for the code listings to be copyable, so the spaces at the beginning of the line need to be be copied along with the text.  This question has been asked in various ways before (e.g. , or ). Those questions focus on making line numbers uncopyable, though.  Making the spaces at the beginning of a line copyable seems to be harder: "I am not sure it is possible to specify in the PDF (at least in a viewer-independent way) that the indentation should be copied too" (). At , Philippe Goutet suggests a solution (turning the spaces into visible spaces, and coloring them in the background color so that they appear invisible) that works using Acrobat Reader, but not all readers. He says "It works under Acrobat Reader and it's extremely pleasant to be able to quickly copy/paste code without problem (perhaps the problem can be circumvented by writing direct PDF code to tell that it's a space, I've never had the time to try)".  Is it possible to produce a PDF with a code listing with copyable real spaces at the beginning of a line?  Minimal example: The line return x should start with four spaces.  \documentclass{article}  \begin{document} \begin{verbatim} def myfunction(x):     return x \end{verbatim} \end{document}   I know that I could attach the code to the PDF as a file, but that's not what I want. Storing hard crusted bread I'm currently baking a batch of sourdough bread. My last batch turned out perfectly with a hard crust and chewy inside. My problem is how to store the bread such that the crust stays hard. If I put it in a plastic bag the crust goes soft in a matter of hours and I'm afraid that the bread will turn dry if I leave it without some kind of protection. Note that I'm not looking for a long term storage solution.  What is the best way to store my bread, such that the crust stays hard and the inside stays chewy? So I have some ISO stock options (yeah) and three kids in college (boo) so I'm trying figure out the best tax strategy. I was hoping someone help me through the following scenario  Strike Price: $10, 2014 exercise price: $20, 2015 price: $25, 2016 price $30   Lets' assume that all of this happens 2+ years after the grant date and that my "normal" tax bill is right on the brink of the AMT threshold, that the AMT tax rate is 28% and the long term capital gains tax is 15%.   Now I want to run the following scenario  2014: exercise 1000 options 2015: exercise 1000 options, sell the previous years' option after "one year and one day" 2016: sell the 2015 options after "one year and a day"    Tax wise that would look like this  2014: AMT on $10000 of bargain element so $2800 AMT 2015: AMT on $15000 bargain element $4200, long term gain tax on $15000 = $2250 2016: long term gain tax on $20000 = $3000    What's the overall tax liability in each year? I can't figure out how the AMT carries forward when the options are finally sold.   Any good pointers to modeling software, spreadsheets or decent papers would also be appreciated. Is an integer a sum of two rational squares iff it is a sum of two integer squares? Let $a\in \mathbb Z$. Is it true that $a$ is a sum of two squares of rational numbers if and only if it is a sum of two squares of integers?  I came to face this problem while dealing with quaternion algebras. Grammar-checking tool for use with LaTeX? I am looking for a grammar-checking tool, i.e., something that checks for mistakes such as "two dog" or "a books".  It has to work on Linux (Ubuntu).  Also, since I use macros to generate some of the prose, it has to somehow work with LaTeX.  I don't think that such checkers are LaTeX-aware, so it has to work with some compilation product of LaTeX: DVI or PDF.    If you suggest a grammar checker that checks plain text, please explain how to you extract the text.  catdvi does a nice job, but it leaves some garbage like page numbers. Does it make any difference for SEO benefits to buy dropped and already available domains other than backorder domains that are expiring soon?  I heard somewhere that whois gets deleted after being dropped. But I didn't find any confirmation if it's true or not.  Anybody can tell me if there are benefits of one against the other, if yes what? Understanding GeoTIFF tags? Edits that shadow a suggested edit inconsistently reject but include the suggestion I have the edit privilege. If I edit a post which has a pending suggested edit, then:  My edit starts from the version resulting from the suggested edit. I'm not notified of that, or even of the existence of the suggested edit. (There's actually a clue, but it's easy to overlook — the edit summary field is pre-filled with the summary from the suggested edit.) The suggested edit is marked as rejected by Community with the message “This edit conflicted with a subsequent edit.” The suggester is not credited in the edit history, my revision combines the suggested edit with my modifications.  This can happen if an edit is suggested while I have the page open: I click on “edit”, and think I'm editing the version that I was just reading, but in fact I'm editing somebody's suggested edits. It also happens if I deliberately visit /posts/12345/edit, which . Examples: a  that I approved and then wanted to ; a  where I . There are two bugs here, but I'm making a single report because they're very closely linked.  I'm not told about the suggestion, so my edit includes content that I'm not aware of, yet is attributed to me. The suggestion is marked as rejected and the suggester is not credited in the edit history, yet the suggested changes are made. iPhone Apps in Waiting mode for the last 4 days I tried to download the "Dictionary Free" app from the App Store four days ago, but it's been stuck in "Waiting" ever since. I've tried everything: rebooting the phone, restoring from an iTunes backup, side-loading the app from iTunes, but nothing has worked.  When I turn Airplane Mode on and then off again, I get an alert popup that says "Unable to download application. It will be available for download when you log in to the iTunes Store on your computer".  I've tried what it recommends, but nothing seems to work. I have to give a presentation on the theorem in Real Analysis with a fellow student. While I've looked over the  and verified that, yes, step B does indeed follow logically from step A, etc. and have internalized the proof to the extent that I can replicate it myself on paper, I still feel that I have made little progress as to why this theorem works the way it does, i.e. what is the intuition behind the theorem, such that it should make sense that it follows the way it does. Therefore I ask: what kind of intuition is there about this theorem? It would also be helpful to understand how the theorem can be used effectively in analysis.  I have seen it referred to as a sort of L'Hopital's rule for sequences, which certainly seems to make sense, but I similarly have little intuitive understanding of how that rule works, either.  I am asking this question not only for my personal understanding, but also for the sake of being able to present it in an illuminating manner, such that the rest of the class can also come away with the same sort of intuition of how the theorem works and how it is useful. Any help would be greatly appreciated.  EDIT: I should point out that the formulation of the theorem we are being tasked to prove is the following:  Let $\lbrace a_n \rbrace$, $\lbrace b_n \rbrace$ be sequences, ${b_n}$ strictly increasing and unbounded. Then if $\lim\limits_{n \to \infty} \frac {a_{n+1} - a_n}{b_{n+1} - b_n} = l$ for some $l \in \mathbb R$, then $\lim\limits_{n \to \infty} \frac{a_n}{b_n} = l$ also. I am using lightning:recordEditForm and lightning:inputField for creating a new record. this component is being used inside a lightning community builder. When i click the submit button, i am able to create a new record in my custom object. However i need the created record ID as i need to pass it in the URL for the next page. Probability of 3 people in a room of 30 having the same birthday I have been looking at the birthday problem (http://en.wikipedia.org/wiki/Birthday_problem) and I am trying to figure out what the probability of 3 people sharing a birthday in a room of 30 people is. (Instead of 2).  I thought I understood the problem but I guess not since I have no idea how to do it with 3.