I know that the minus sign of -r, where r is real, can never be moved under a radical sign. But there are no minus signs present when the sqrt of $i^4$ is taken in the following manner  $1 = i^4$  $\sqrt{1} = \sqrt{i^4}$  $+/-1  =  +/-i^2$  $+/-1  =  -/+1$  I realize it might be argued that i also can never be moved under the radical sign.  But $\sqrt{i}$ does have an open form solution which is $+/-[(1+i)/\sqrt2]$.  Repeating the above calculation with this open form I find that the associative law of multiplication no longer holds and all sorts of contradictions ensue. It might also be argued that $i^4$ factors into -1 x -1 and therefore cannot be placed under the radical sign. But that is also true of any positive real number. Any ideas for a consistent solution to the above derivation?  Thanks.
What is the $\sqrt{i^4}$?   $i^4$ = $(i^2)^2$ So is $\sqrt{i^4}$ = $\sqrt{(i^2)^2}$ = $i^2$ = $-1$?  Or is $\sqrt{i^4}$ = $\sqrt{1}$ = $1$?  When I plug it into my TI-89 Titanium, I get $1$.  Edit: I now see why $i^4 \neq -1$. $$\sqrt{x^2} = \left|{x}\right| $$ So$$\sqrt{(i^2)^2} = \left|i^2\right| = \left|-1\right| = 1$$
Asking users to accept answer when they comment that it answers their question I've had a couple of answers today where the OP indicated via a comment that I solved their problem, but the question wasn't upvoted or accepted.  In both cases the person probably wasn't able to upvote since they were new, but they could have accepted my answer. I felt a little guilty asking them to accept the answer, but since it solved their problem I felt that it would help others who stumbled on the question trying to solve their own problem to know that my solution really was the answer. Is asking the OP to accept your answer when they've said that it solved their problem a reasonable thing to do or does that feel like nagging for reputation? How can I extract the points from a ListSurfacePlot3D? Windows 10 Search not loading, showing blank window For any smooth manifold, is it true that for any two points on the manifold, there exists a chart that covers the two points? Some say that we can connect two points with a continuous curve and a small contractible neighborhood of the path together with the charts of the points can be regarded as the chart. But I don't know how to verify this statement. Improving Review Queues - Design overview II: Changing review bans and other new features This is one of three project announcements for Improving Review Queues. We’ve summarized the project objectives and goals . As a reminder, this project is still in the . In this post, we are sharing proposed changes to review bans and other new features. We are asking for your feedback before we begin implementation. After we collect community feedback, we will be open to including changes into the next design iteration. Suspension of privileges  Most users learn how to be good reviewers by actually reviewing posts, but sometimes find themselves making wrong decisions. This can lead to a less than obvious suspension of privileges, with too little guidance as to what they did wrong and how they can improve in the future. In the event of a suspension, a user will be notified in the Review Queue dropdown. A post notification will be available with reason and time remaining on the Review Queue main page, along with guidance for continued learning in this area. We’re also proposing softening the language by renaming “review bans” to “review suspensions.” Open questions:  Do you have suggestions of other ways we can notify reviewers of a ban? What information would be most helpful to you in this situation?  New features  My tasks We’re considering introducing a new, additional workflow and means of task discovery. My Tasks is a curated experience that’s currently based on your Watched Tags. This page includes an overview of pending tasks that may require more immediate attention or pique interest in a user’s subject-matter expertise.  Queue filtering We’d also like to add more useful, robust filtering options on all queues. Right now, the filter function is difficult to discover. We want to make this feature more noticeable and add queue-specific options. For example, you will be able to filter by your Watched Tags and sort tasks that are soon to expire out of each queue. Badges Currently, the gold Steward badge is awarded once at 1,000 reviews per queue regardless of whether you’ve performed 1,000 or 10,000 reviews. We propose that reviewers can now earn this badge multiple times, adding more opportunities to earn badges while reviewing - this change is intended to be retroactive. Open questions:  How would you want to be able to customize in My Tasks? Are there other review-centric badges you would like to see added?  In case you missed it... We propose changes to Onboarding and Updated workflows/pathways here: . How do I resolve these unmet dependencies? effect of mass of spring on period of oscillation Consider a massless spring system which is hanging vertically. Attach a mass $M$ and set it into simple harmonic motion. Let the period with which the mass oscillates be $T$.   We assume that the spring is massless in most cases. Would taking effect of the non-zero mass of the spring affect the time period ($T$)? ActionBarSherlock + Google Maps API v2 Duplicate Id What are the badges I can earn on each site, and what are the exact criteria for earning each badge? closed form for $\binom{n}{0}+\binom{n}{3}+\binom{n}{6}+...+\binom{n}{n}$ How can I add a PPA from a previous release? How can I prove the convergence of the sequence $b_n=\sqrt[n]{x^n+y^n}$ where $x, y &gt; 0$?  Can I divide it in two cases?  Case 1: $x &gt; y$.  $$ b_n=\sqrt[n]{x^n+y^n} &lt; \sqrt[n]{x^n+x^n} = \sqrt[n]{2 \cdot x^n}=x \cdot \sqrt[n]{2} $$  Case 2: $x &lt; y$.  $$ b_n=\sqrt[n]{x^n+y^n} &lt; \sqrt[n]{y^n+y^n} = \sqrt[n]{2 \cdot y^n}=y \cdot \sqrt[n]{2}$$  Result: Does the sequence converges to $\max(x,y)$? How to do a secure erase on a Macbook Pro? I have a Macbook Pro with SSD and I would like to completely erase every single bits of data on it. How would I do that? On a PC, I would just run Darik's Boot And Nuke, but can I run it on a Mac too, or is there a better alternative? We currently have about 50 pages, each of which I want a user (eg, bob, rob, smith) to be able to edit only 1 page.  For example, bob &amp; smith each have their own page.  I do not want bob to be able to edit smith's page.  I want bob to ONLY be able to edit bob's page.  I don't care if he can see other pages. Looking through the user roles, I don't see a way to currently to do this- I only see how to allow restrict access on a global scale.  Are their any plugins to help restrict edit access per user, or roles restricted to editing specific pages and I can just add 1 user per role?  Or was there a way to do this with default settings I missed. Why there isn't a feature that lets any user on SO to request other user to answer their question? When I browse through other questions to find my answer, sometimes I'll find answers similar to what I'm looking for, and though they don't exactly answer my question, the author might be able to do so. Could we have a feature on SO that will let me request that another user answer my question, in case I find someone who could potentially answer my question? Need help with proof for Dedekind cuts on $\mathbb{Q}^+$ I am working on a proof about Dedekind cuts on the positive rational numbers. I have been stuck for a while on the following point and would appreciate any help.  Given $x\in \mathbb{Q}^+$ such that $x^2&lt;2$, how to find $y\in \mathbb{Q}^+$ such $y&gt;x$ and $y^2&lt;2$? How to preserve bookmarks when rearranging pages of a PDF file with tools like pdftk? I am using pdftk to rearrange pages of a pdf file with bookmarks/outlines, but after that, the output file lost the bookmarks of the original.   My command is pdftk in.pdf cat 1-22 43 23-42 44-end output out.pdf.  I was wondering how I could keep the bookmarks while rearranging pages? I have a nominal variable (different topics of conversation, coded as topic0=0 etc) and a number of scale variables (DV) such as the length of a conversation.   How can I derive correlations between the nominal and scale variables? Not enough storage I have an 8GB iPhone. I've had not enough storage for a while so I went through my photos and deleted 100. I also deleted old notes that I don't need. I also deleted an app of 81.5 MB. I tried downloading an app of 22.1 MB but it's telling me I don't have enough storage. Is this a bug or do I just not have enough storage? Difference between (function(){})(); and function(){}(); Possible Duplicate:          This is something I haven't quite figured out yet, but I have been using function(){}() just because my VIM syntax highlight screws up if I add the parenthesis, although I've seen (function(){})() around many times, maybe its an IE thing?  edit:  var singleton = function() {     // code }();  var singleton = (function() {     // code })();