Can anyone please check my proof for this number theory problem?

Math100
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Homework Statement
Prove that ((a, b), b)=(a, b).
Relevant Equations
None.
Please view the picture of my work which I've uploaded.
 

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Looks good.
 
PeroK said:
Looks good.
Thank you.
 
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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