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I have the arguments and modulus of some values of $^{1/},$ where $$ is a complex number and $n$ is a positive integer. The values of $x^{1/n}$ I have are called $x_1,x_2,$ and $x_3.$ I want to find the smallest possible value of $n,$ and also use that value of $n$ to find the value of $x.$

$\arg(x_1) = \frac{7\pi}{18},\quad \arg(x_2) = \frac{5\pi}{9},$ and $\arg(x_3) = \frac{31\pi}{36}.$

The modulus of $x_1,$ $x_2,$ and $x_3$ are all $2.$ In other words, $|x_1| = |x_2| = |x_3| = 2.$

I understand that the $n$ $n^{th}$ roots of a complex number have arguments spaced by $\frac{2\pi}{n}.$ So I understand how to find the smallest possible value of $n$ if we are given two values of $x^{1/n},$ as the roots all differ by an integer multiple of $\frac{2\pi}{n}.$

This is what I have tried so far:

  1. I found $\arg(x_2) - \arg(x_1) = \frac{\pi}{6}$ and $\arg(x_3) - \arg(x_2) = \frac{11\pi}{36}.$
  2. For $\frac{\pi}{6}$, $n$ must be an integer multiple of $12$ to get an integer multiple of $\frac{2\pi}{n}.$ For $\frac{11\pi}{36},$ $n$ must be an integer multiple of $72$ to get an integer multiple of $\frac{2\pi}{n}.$

But I am not sure where to go from here. Am I correct in thinking that $72 = n?$ How would I use this to find the value of $x?$

user376343
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avocado_pit
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1 Answers1

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You have already proved that $72$ is the smallest value convenient for $n.$ Further,$$\frac{7\pi}{18}=14\times\frac{2\pi}{72},\quad\frac{5\pi}{9}=20\times\frac{2\pi}{72}\quad \text{and}\quad \frac{31\pi}{36}=31\times\frac{2\pi}{72}.$$ The set $\{14,20,31,72\}$ is coprime. We deduce that the number $x$ is real, positive (hence $\arg(x)=0$), and so $$x=|x|=|x_i|^{72}=2^{72}.$$

user376343
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