The iteration of exponents laws I learned gives: $$i^{4/3}=i^{4\cdot1/3}=(i^{4})^{1/3}=1^{1/3}=1$$ I know this is wrong but now the iteration of exponents laws I have used over the years somehow doesn't apply or is flawed. This is quite embarrassing as I have been studying mathematics for quite some time now. Please help.
I know there is something wrong with this but I don't know where. It's some kind of a math fallacy and it is driving me crazy. Here it is: $$-1= (-1)^3 = (-1)^{6/2} = \sqrt{(-1)^6}= 1?$$
Increasing post view count automatically Initializing C# auto-properties I'm used to writing classes like this:  public class foo {   private string mBar = "bar";   public string Bar {     get { return mBar; }     set { mBar = value; }   }   //... other methods, no constructor ... }   Converting Bar to an auto-property seems convenient and concise, but how can I retain the initialization without adding a constructor and putting the initialization in there?  public class foo2theRevengeOfFoo {   //private string mBar = "bar";   public string Bar { get; set; }   //... other methods, no constructor ...   //behavior has changed. }   You could see that adding a constructor isn't inline with the effort savings I'm supposed to be getting from auto-properties.  Something like this would make more sense to me:  public string Bar { get; set; } = "bar"; I'm finishing professional school and over the years have acquired a pile of notes and articles that I'd like to hang onto.  I'd like to add to them and create sort of an archive of article and files that may be useful down the road.  I'd also like to organize this collection of files not only by simple grouping but also with tags.  I feel like that will make searching through them years later much easier.  Suggestions on software that would be good for this?  Just a general file manager, something that uses tags that can be attached to each file? If $S$ is a finitely generated graded algebra over $S_0$, $S_{(f)}$ is finitely generated algebra over $S_0$? Let $S = \bigoplus_{n\ge 0} S_n$ be a graded commutative ring. Let $f$ be a homogeneous element of $S$ of degree $&gt; 0$. Let $S_{(f)}$ be the degree $0$ part of the graded ring $S_f$, where $S_f$ is the localization with respect to the multiplicative set $\{1, f, f^2,\dots\}$. Suppose $S$ is finitely generated algebra over $S_0$. Then $S_{(f)}$ is a finitely generated algebra over $S_0$. Why put a resistor in series with signal line? A lot of times in circuits I see a resistor placed in series in a signal line and sometimes even in series with an MCU's VDD line. Is the intention of this to smooth out noise in the line? How is this different from using a small cap, like a .1µF to do the same thing? Why does kworker cpu usage get so high? I recently installed Ubuntu 12.04 on my Samsung laptop alongside Windows 7. Whenever I use Ubuntu (even when its idle) kworker uses almost 90% of one of 8-cores. Even though it doesn't really affect my usage, it bugs me and I am afraid of that it might harm my processor. I even tried installing another Linux distro (Linux Mint), and kworker caused the same problem. So I don't know what to do. I would really appreciate your  help. Quantum systems without a classical analogue? I am now reading the quantum mechanics textbook by Dirac (chap. 4, $\S21$, p. 88).   He says that his quantization procedure does not include all possible systems in quantum mechanics and there are systems without a classical analogue.  How is this possible? Load your bag or Load your bike? Probably, another debatable question from my side.  When carrying items while riding, like water bottle, repair kits, pumps, or other stuffs that you might need, What is the best option.... to carry them?   On your cycle at saddle bag, handlebar bags, or pannier bags or etc.  or, On backpacks, like hydrant pack, or simple travel pack.   While loading on cycle, it will increase the weight of the bike, thus affect a lot of things, like speed, traction, more reactive on corners. Like while using pannier bags.. it will increase rear weight, making it difficult on uphill.   Also while loading on backpack, it will tire you down fast if its a big load, decreases your flexibility, agility &amp; etc.  So what is right way to do it?  Update:  The type of ride I generally, go to are commuting, mountain biking, &amp; trailing. And the items I generally carry are   Commuting: Water bottle(x2), Laptop (17"), books, copies, repair kits, hand pump, few other accessories. Mountain Biking &amp; Trailing: Water bottle(x2), repair kits, extra tubes, hand pumps, candybars &amp; few other accessories Memory and CPU optimizers: Are they gimmicks? How can I recover contact information from a stolen phone? Did regular Americans speak the way actors in the 30s and 1940s did? I watch a lot of old movies, and I've noticed that American actors of the 1930s and 1940s often spoke in a quasi-generic-posh-British accent.  Katherine Hepburn's accent would be the perfect example. It seems exaggerated, and I imagine it was not common off the stage and screen. What was that? Was it just an accent or was it a dialect? Did real people actually speak that way? Why use sucrose rather than glucose in an injectable vaccine? Proof conclusion to: Let m,n $\in$ $\mathbb{N}$ $\backslash$ $\{0\}$ . Prove that $s |\tau (m^{s}) - \tau(n^{s})$, where $\tau$ is a divisor function.   Hello, so I have a tiny question about the summarization of my proof.  I got to the point where I defined my $\tau$ divisor function and got them both $\tau(m^s)$= $\prod$ something +1 and $\tau(n^{s})$ = $\prod$ something +1.    I've noticed that the last terms of the polynomial are both one. Why is it with the +1 part, I can conclude with that:  s| $\tau(m^s)$ - $\tau(n^s)$.   The line for why that is, isn't quite clear. Before asking my question, I'll use an example to clarify the idea.  Say that a company used to sell horses and then they turn into a sheeps business. While some of the old pages can be mapped to the new pages using a 301 redirect, because they're really related in the content and the company don't want to lose users and link juice, there are a lot of pages strictly related to horses that they want to shutdown.  What's the best practice? Is there a way to say to users and Search Engines the content you were looking for won't be available any more?  Basically a 301 redirect of the closed pages to new ones seems not so good, because you'll show a content totally unrelated to the searched one.  P.S: I'm aware that probably the best fallback is to simply map those pages to the homepage but I hope in a smarter solution. Setting CSS pseudo-class rules from JavaScript Does "until [date]" mean "before that date"? What does until mean in the following?     You need to deliver this product within 2 days (until August 18, 2011)   to meet your deadline and get paid.   Does this mean that I have to deliver the product by 23:59 on 17 August 2011 or 23:59 on 18 August 2011? The order time is 21:04 on August 16 2011. Kernel did not upgrade between 14.04 LTS to 16.04 LTS Eclipse Scala IDE - class file needed by X is missing It's often the case that I want to change name servers because my site is messed up.  Well, it takes a very long time for the name server to propagate.  Is there something in DNS, or registrar or anything I can do to speed that up.  Flushing my dns server using  and it rarely works Can a finite sum of square roots be an integer? Can a sum of a finite number of square roots of integers be an integer? If yes can a sum of two square roots of integers be an integer?  The square roots need to be irrational.