Understanding the Product Rule: Common Factor Simplification Confusion Explained

fitz_calc
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I've already done the problem (here is the last part, sorry for crappy pic)

Picture002.jpg


My question; if I multiply both sides of the + by x^2(x+1) / x^2(x+1) as the common factor to simplify the equation, then why does on of the x^2(x+1) disappear from the numerator in the final answer? It appears as though both denominators were used to simplify both sides of the +, yet only one numerator was used in the final calculation. Please tell me I'm not going crazy!
 
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Because the two terms in []'s on the first line are combined into a single [] on the second line.
 

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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