Why is $(e^{2πi})^i$ different from $(e^i)^{2πi}$ I'm a high school senior and this question has been on my mind since 8th grade, since I've learned Euler's formula but I've thought about it a lot more recently. According to Wolfram Alpha:   How is this possible?????? In other words, if $$ e^{iπ} = -1 $$ $$ e^{2iπ} = 1 $$ $$ (e^{2iπ})^i = 1^i $$ $$ e^{-2π} = 1 $$ Where is the error in this reasoning?
What is wrong with this fake proof $e^i = 1$? $$e^{i} = e^{i2\pi/2\pi} = (e^{2\pi i})^{1/(2\pi)} = 1^{1/(2\pi )} = 1$$  Obviously, one of my algebraic manipulations is not valid.
I tried to upgrade from Ubuntu 16.04 to Ubuntu 18.04. First, after downloading and installing of packages, I was unable to login. Login screen hanged completely and it showed Failed to authenticate. But then I logged into tty1 and then logged in Ubuntu. It was saying that I am in Ubuntu 18.04. Then it asked for reboot. So I rebooted Ubuntu from terminal. It got stuck at something like /dev/sda4 : Cleaning files. After trying for a few times, I got into tty1 and here I logged in as root user and tried to update and upgrade. But now I am getting errors like unmet dependencies and broken packages. I have done sudo dpkg -- configure -a. I don't want to do a reinstall of Ubuntu. How do I get access to my Ubuntu Desktop? What is the best way to modify a list in a 'foreach' loop? Doubles rules in table tennis How do I post code in chat? I would like to be able to post code in chat. I know that I can post inline snippets of code by using backticks, but I don't know how to post blocks of code, with the entire chat message formatted as code, and indentation respected. I am aware of . After reading the comments, I found out that I could format in "fixed width", but I can't see a "fixed width" button anywhere. What should I do? When there are X New Questions, the views don't update untill you refresh the page How can I disable landscape mode in Android? In the beginning of the series, it's mentioned that Muggle-born wizards/witches are identified due to inadvertent spell-casting. However, it seems like later in the book, disarmed wizards/witches are completely helpless.   What is the explanation for why some spells can be cast without a wand and is there an in-universe explanation for why this isn't done? Also, are there spells that can be cast non-verbally (with/without a wand)? arccot limit: $\sum_{r=1}^{\infty}\cot ^{-1}(r^2+\frac{3}{4})$ Suppose $\sqrt6 = \frac pq$ where $p$ and $q$ have no common factors.  $$6 = \frac {p^2}{q^2}$$  $$6p^2 = q^2$$  So $q^2$ and therefore $q$ is divisible by $6$.   $$p^2 = \frac {q^2}{6}$$  So $p^2$ and therefore $p$ is divisible by $6$.   S0, $p$ and $q$ have a common factor $6$. Contradiction.  Therefore $\sqrt6$ is irrational.  Does the proof work? Why shouldn't we use words such as 'here' and 'this' in textlinks? Over the years of surfing on the web I have found numerous examples of people saying that you shouldn't use the word "here" in a textlink etc. Personally I like the idea of using the word, because it is so straightforward for your visitors to know where to click.  The only reason that I can truly think of is for SEO purposes. The word here doesn't have much value as a keyword as such.  I would like to know some other opinions on why we should/shouldn't use word like "this" and "here" in a textlink. Proving that $a \mathbb{Z} \cap b\mathbb{Z} = \operatorname{lcm}(a,b)\mathbb Z$ Given $m \in \mathbb{Z}$, let $m\mathbb{Z}$ denote the set of integer multiples of $m$, i.e. $m\mathbb{Z} := \{mk\mid k \in \mathbb{Z}\}$.   Now let $a,b \in \mathbb{Z}$ with $a,b$ not both $0$. Prove that $a\mathbb{Z} \cap b\mathbb{Z} = \operatorname{lcm}(a,b)\mathbb{Z}$.   I am trying to write a proof for this, but I am unsure of what method to use. Also I am confused by $mk\mid k$, because wouldn't $m=1$ for this to be true. Running junit tests in parallel in a Maven build? How to mount EXT4 disk on 10.15? Since I have osxfuse installed it would be nice if I could use it. There are a  (heavily) outdated  here, saying that it is not . Is this still the case? Is there any way to increase the drop rate of items? Mathematica Sec and Csc I followed the instructions and got success response but I didn't receive email.  Then I used the id to get Email Status and got 404 error.  Here is my json :  POST :   {     "To": {         "Address": "example@gmail.com",         "SubscriberKey": "example@gmail.com.tw"     },     "Options": {         "RequestType": "ASYNC"     } }   GET : {id}  Please let me know how I can resolve the issue. How should I phrase a question that must be answered with an ordinal number (e.g., the third prime)? How do I disable the volume control overlay? When changing the volume on a Mac, you get a big ugly gray bezel showing the change in volume. This is super annoying when (say) watching a movie. Is there any way to disable this? What is a tty, and how do I access a tty? I'm putting some type on a path, a circle; but when I do it, the type is upside down. How can I flip it so it reads the right way up? See image below: