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Consider the complex numbers  in the complex plane below:

Some of these complex numbers are 54th roots of unity. Enter those complex numbers in any order, separated by commas. Use the letters to refer to the complex numbers; do not write them in or form.

 Apr 3, 2022
 #1
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The 54th roots of unity are u, v, and w.

 Apr 3, 2022
 #2
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I think just v

 

\(\frac{360}{54}=6.\dot6\; degrees \)

 

 

So there will be a 54th root of 1 at 1+0i    

then there will be one every 6.666.. degrees around the unit circle.

 

That graph is split into 24 sectors   360/24 = 15 degrees each

 

so we need integers such that 15x=6.666.. y     where   x is between 0 and 24 and x and y are both integers

 

The only one that works is 180 degrees  

15*12 = 27*6.6666  

So the only answer is v

 Apr 4, 2022

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