I was reading Richard Feynman biography when I read that one time he was able to calculate the  cube root of large number in his brain by just using simple facts of everyday life. So my question is there a way to calculate square root or cube root by using just a pen a paper, I mean a systematic way not by trial and error.
Suppose that we don't know logarithm, then how we would able to calculate $\sqrt x$, where $x$ is a real number? More generally, is there any algorithm to calculate $\sqrt [ n ]{ x } $ without using logarithm? More simple techniques would be nice.  Here is a simple technique used to approximate square roots by :  For example: $\sqrt {78}\approx 8\frac { 14 }{ 17 } $ , where $8$ is the nearest integer root of $78$, $14 = 78 - 8^2$, $17 = 2 \times 8 + 1$.  if $n=2^k$ we can use the method above.   For example, for $k=2$ Lets calculate $\sqrt [ 4 ]{ 136 } $: $$\sqrt [ 4 ]{ 136 } =\sqrt { \sqrt { 136 }  } \approx \sqrt { 11\frac { 136-{ 11 }^{ 2 } }{ 11\times 2+1 }  } =\sqrt { 11\frac { 15 }{ 23 }  } \\ \sqrt { 11\frac { 15 }{ 23 }  } \approx 3\frac { 11\frac { 15 }{ 23 } -{ 3 }^{ 2 } }{ 3\times 2+1 } =\frac { 544 }{ 161 } =3.38\\$$ The exact result is$$ \sqrt [ 4 ]{ 136 } =3.4149\cdots$$ The method approximates well, but it is working for only $n=2^k$ as I know.
Is Java viable for serious game development? Performing test for homogeneity of variance without samples I'd like to perform a test for homogeneity of variance in r, but I don't have samples available to test.    I have the sample mean of each group, the sample deviation of each group, and the sample size of each group(100).  Brand    ybar_i  s_i    n_i --------------------------- LowTar   9.64    .291   100 A        10.22   .478   100 B        10.77   .372   100 C        11.57   .352   100 D        13.59   .469   100   I'm thinking I should use the Levene test from Lawstat package, but that requires samples to test.  Or I think I could use the Fmax test. but that also requires samples. What is the most efficient way to make money? What are some of the most efficient ways to make money?   Based on:    time initial investment required progress (in the game) special equipment/gear Is my proof of convergence for $\sum\limits^{\infty}_{n=1} \frac {2^n} {n^n}$ true? I want to prove that $\sum\limits^{\infty}_{n=1} \frac {2^n} {n^n}$ is convergent.  Therefore I want to use that it will be convergent if $\lim\limits_{n \to \infty}|\frac {2^{n+1}} {n^n}| &lt; 1$ is true.  But I am not sure if all of my steps are legal to do in order to do it. Here is what I came up with:  $$\lim_{n \to \infty}|\frac {2^{n+1}} {n^n}| &lt; 1$$ $$\lim_{n \to \infty}|\frac {\sqrt[n]{2^{n}} \sqrt[n]{2}} {\sqrt[n]{n^n}}| &lt; \sqrt[n]1$$ $$\lim_{n \to \infty} \frac {2 \sqrt[n]2} {n} &lt; 1$$ $$0&lt;1$$  $\Rightarrow$ It is true. I've been thinking about switching to Ubuntu full time, but there are still some programs left behind that I haven't found any equivalent to yet. Does anyone know of a music player similar to Foobar2000 (I've read it doesn't handle well in WINE)? Overflow: Visible on SVG If I buy a game from PSN on ps3 Is it permanently mine? What part of speech is "back" in "put the book back on the table"? Put the book back on the table.   I'm having trouble. I think it is a preposition. Do I need an ESTA to enter the US with the Victoria Clipper ferry? I am thinking of visiting Victoria, BC in Canada and then going to Seattle, WA in the US, with the .  I am a national of a country under the Visa Waiver Program. I think that if I enter by air, I need to fill an ESTA form. If I enter by car, I don't need one, I can fill a form when crossing the border.  With the ferry, what is the procedure? Should I necessarily have an ESTA? Convert and redirect URL in uppercase to lowercase using .htaccess I would like to redirect a URL with uppercase characters http://localhost/A/B to a lowercase version http://localhost/a/b using the .htaccess file. Using regex I can capture A and B but is it possible to convert them to lowercase? Problems with Curve Modifier Every time I think I have the curve modifier mastered, it throws me a another curve ball (no pun intended).  Why in  can I not get the lights to follow the spiral curve? When the modifier is activated, the lights are stretched and distorted along the curve.  I created the curve by converting it from a mesh (string of verts). Why does it not display like a normal Bezier? Is that related to my problem?  I am following all the "rules" that I am aware of, as detailed here: . The curve seems to run in the correct direction. The origin point is at the first point of the curve. The curve modifier on the lights is set to -Y which is the local orientation of the curve's direction.  What am I missing? I was trying to dual boot elementary os with windows on my laptop (Asus Zenbook UX305). To do that I shrank the windows partition using the windows disk manager, and then used the Elementary installer's "install alongside windows" option to set up the linux partitions automatically. When I rebooted after installing I was put into the grub console. I was able to boot linux manually with the commands  grub&gt; set root=(hd0,gpt4) grub&gt; linux /boot/vmlinuz-3.19.0-26-generic root=/dev/sda4 grub&gt; initrd /boot/initrd.img-3.19.0-26-generic grub&gt; boot     and I found that running the command  grub&gt; configfile (hd0,gpt4)/boot/grub/grub.cfg   would load the normal grub menu as is supposed to happen - so there don't seem to be any problems with the config file itself. I also found that the root and prefix are on (hd0,gpt1), so running  grub&gt; set root=(hd0,gpt4) grub&gt; set prefix=(hd0,gpt4)/boot/grub grub&gt; insmod normal grub&gt; normal   also loaded the normal grub menu. So possibly the problem is that grub is looking on the wrong partition?  In my BIOS, I have a choice between using grub and the windows bootloader to boot. Selecting the windows bootloader boots me directly into windows without any problems.  However, I can't find any way to get any fixes to stick after a reboot. I've tried rerunning grub-install /dev/sda, recreating the config file with grub-mkconfig, and using the Ubuntu boot repair gui tool - none of these have worked. Is there any way to fix this short of reinstalling either or both OSs? I'd rather not have to reinstall windows if possible. Ridiculous error on all google play store Using CM 10.1.3 interaction between mathematical structures From a physicist's perspective there are several situations in which somehow arbitrary choices of mathematical structures can be made. One can describe a system from different perspectives, etc. without changing anything in the physical properties. (see gauge invariance, picture changing etc.) In the situation when these choices impose a mathematical structure one has limitations in the subsequent choices (are these limitations physical???). However, there are situations when this prescription does not hold. The best example is in this case the Black-Hole complementarity principle (if it is well defined in the first place). So, the question about how mathematical structures "interact" in a logical way is not a bad one. Again: what is the interaction between, say, locality and Hausdorff-ity in the case of a black hole? Or what is the true relation between metrizability and hausdorff-ity of a space in the context of a black hole? The question about how mathematical structures assigned (innocently) at some point interact in different situations is in my opinion a quite relevant one... When Bottom Navigation Isn't Best I'm developing a mobile app and dealing with various UI/UX issues. I'm open to all kinds of ideas at this stage, however some designers I consult with seem to be zealots for bottom-navigation and are hesitant in actually analyzing whether the bottom-nav is actually best for my app; they just want me to use the bottom tabs navigation as if it's some UI law.  One point I try to bring up is that my business model has users who aren't typical one-handed mobile users, are tech savvy, would use my app more of a tool, and would spend only a few minutes in it. The app isn't a typical lazy-content-browsing experience where you want to minimize fingers-over-top-of-screen interference, and where you want to keep the use transfixed and spending time in the app.  Anyways, I'd like to hear about situations where a bottom nav was decided-on, or started-with, and you had good reason to eliminate it and go with another navigation design.  Most articles out there talk about eliminating hamburger menus and diving into bottom nav. Is the converse simply going back to hamburger menu? Why does an unhinged body rotate about its centre of mass? C#: code error while changing the active directory user's password Changing Locale within the app itself What is the default password of the sudo? I visit some forums, it tells that the default password is password, but it doesn't work, how can I determine the password? Strip whitespace from jsp output