$$(z+1)^6=z^6 $$ $$\frac{(z+1)^6}{z^6 }=1$$ $$\left(\frac{(z+1)}{z }\right)^6=1$$ $$\left(\frac{(z+1)}{z }\right)^6=\cos(2*\pi*k)+i\sin(2*\pi*k)$$ $$\frac{(z+1)}{z }=\left(\cos(2*\pi*k)+i\sin(2*\pi*k)\right)^{1\over6}$$ $$1+\frac{1}{z }=\cos({2*\pi*k\over6})+i\sin({2*\pi*k\over6})$$  $$\frac{1}{z }=(\cos({\pi*k\over3})-1)+i\sin({\pi*k\over3})$$ $$\frac{1}{(\cos({\pi*k\over3})-1)+i\sin({\pi*k\over3}) }=z$$ I dont know how else to solve this. A lot of help is appreciated. Also i know that k should go from 1-5.
Find all $x$ such that $$x^6=(x+1)^6.$$   So far, I have found the real solution $x= -\frac{1}{2}$, and the complex solution $x = -\sqrt[3]{-1}$.  Are there more, and if so what/how would be the most efficient find all the solutions to this problem? I am struggling to find the rest of the solutions.
Which is the best way to put function plots into a LaTeX document? Should I use  tag for icons instead of ? How to choose a circle for default stream in Google Plus? How can I choose a specific circle for the default stream in Google+ ? Many extensions don't work, and are not comfortable if using another PC. How to pick one number from each sorted list so that the difference between the minimum and maximum is as small as possible? StandardSetController with History records Magnetic Field in the presence of a conductor I am studying for my quals and came across an old question that reads like the following:      There are two regions in space separated by an infinite conducting plane. Region 1 has a magnetic dipole of moment $\vec{\mu}$ located a distance $d$ from the plane. What is the magnetic field $\vec{B}$ everywhere in region 1?   I know that for a perfect conductor (superconductor), the normal component of $\vec{B}$ is zero–although I don't exactly know why–but I don't know if that is the case when the problem involves a "conductor" (not explicitly a perfect conductor).  TL;DR: Can anybody give me some insight about the magnetic boundary conditions of a stationary magnetic field in the presence of a conductor?  Thanks! Are there any simple rules for choosing the definite vs. indefinite (vs. none) article? LSTM : shape of tensors? I’ve recently purchased a new Mac Mini, and I’ve been importing my old pictures into the Photos app.  I want to have these photos backed up to iCloud, along with the photos currently on my iPad. I then want everything synced to my Mac Mini, but not to my iPad, which has limited storage capacity.  Is it possible to have the Mac Mini and iPad set up to backup to iCloud, but only download backed-up photos to the desktop? Weird display on screen before swipe lock Samsung Galaxy s6 edge plus What happens if light/particles exceeded the speed of light for a particular medium? Let $f: \mathbb N \rightarrow \mathbb N$ are increasing function such that $f\left(f(n)\right)=3n$. Find $f(2017)$ Keep values selected after form submission Can open sets and closed sets of $\mathbb{R}^n$ be homeomorphic? Suppose $A$ is an open subset of $\mathbb{R}^n$ considered as a topological subspace. Suppose $B$ is a another susbet of $\mathbb{R}^n$, and under the induced topology, is homeomorphic to $A$.  If $B$ is closed and bounded, then by the Heine-Borel theorem, $B$ is compact as a topological space so $A$ and $B$ cannot be homeomorhpic, because $A$ is not compact. But if $B$ is not bounded, what can be said about $B$, if any? Must $B$ be open, can it ever be closed (let's leave out the only case of being both open and closed--the entire ambient space $\mathbb{R}^n$), or can it be neither open nor closed? 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If I have example.com, and blog.example.com, do search engines see those as one site or two? Will using subdomains hurt SEO? I have some technical reasons why I can't use example.com/blog. Why is "what can I ask about here" not the first entry in the help center? On a lot of sites, the first section of the Help Center looks like this:    When I go to a site's help center it's almost always to find out what's on-topic there, because I'm considering asking or migrating a question.  Leading with what not to ask seems odd to me, and I suspect that to someone not already familiar with SE it could be a little off-putting.  It's not always like this, though; here on MSE we get this, which I've seen on some other sites too:    That ordering makes a lot more sense to me.  It's also consistent with the tour.  (Maybe "how do I ask a good question?" should also be "above the fold", in the top five somewhere.)  Why the difference?  Can we either make the order consistent across the network and more logical, or give users or moderators the ability to change the order on their sites? Can I set background image and opacity in the same property? How to disable USB storage device support I want to disable a particular USB storage device (a sandisk flash drive). I read somewhere that you can use the .config file. How do i use it to disable that?