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We all know that $x^2+y^2=r^2$ is a circle. What does $(x^2+y^2)^2$ signify? In general, what is $(x^2+y^2)^n$?

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  • $\begingroup$ It remains a circle... assuming that $x,y\in\mathbb{R}$. The reason behind it is simply given by the fact that you can always take the n-th root on both sides and you don't change the possibile values... $\endgroup$ – b00n heT Jan 25 '14 at 14:08
  • $\begingroup$ This doesn't answer your question, but it's interesting to note that if $x^2 + y^2 - r^2 = 0$ is squared on both sides, then the resulting polynomial still determines a circle, but all the points are multiplicity two roots of the equation now. $\endgroup$ – Nick Jan 25 '14 at 14:09
  • $\begingroup$ Perhaps you're thinking of the "superellipses", a.k.a. "(even-degree) Fermat curves" $x^{2n} + y^{2n} = 1$, which become "successively more square" as $n \to \infty$...? $\endgroup$ – Andrew D. Hwang Jan 25 '14 at 14:56
  • $\begingroup$ @ hAcKnRoCk : Sorry I cannot understand your question because on one hand you write x²+y²=r² which is an equation and on the other hand (x²+y²)² which is not an equation (there is no = in it). What exactly do you want to compare ? $\endgroup$ – JJacquelin Jan 25 '14 at 15:12
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    $\begingroup$ Higher dimensions = more variables. $\endgroup$ – Martín-Blas Pérez Pinilla Jan 25 '14 at 15:59
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Using the transformation: $$x=r\cos(t),~~y=r\sin(t)$$ there is no change. I mean the circle remains unchanged.

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  • $\begingroup$ Thanks. But i am asking only about the LHS. I have edited the question to make it more clear $\endgroup$ – hAcKnRoCk Jan 26 '14 at 10:39

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