Please explain the solution (Real Analysis)

phillyolly
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What parts do you get? What parts are you having trouble with?
 
My understanding shuts down in the second paragraph, second sentence.
 
Do you understand why \textbf{N}^k is countable?
 

Thread 'Finding the nth roots of a complex number'

There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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