My simple questions - bear with me while I try to learn calculus

[Tim67]

I've just started trying to learn calculus via self-study; I bought a textbook and am using some stuff online for further explanation/practice questions, and if no one minds, I though I might try to use this thread to look for help/advice when I come to a problem I don't understand.

Right now I'm having some trouble understanding this solution:
http://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/limconsoldirectory/LimConSol.html#SOLUTION 7

First off I don't understand how they went from the second centered-line part of the solution to the third - how did (x - 1) come from (x^1/3 - 1)([x^1/3]^2 ... etc)

Also, in the fourth centered line of the solution (i hope my wording isn't confusing anyone -- don't know how else to refer to it), I don't understand how that conjugate was derived.

Thanks for any help that can be given.

 

[zooxanthellae]

Here is the top part of that step 2 - step 3:

(x^(1/3) - 1)((x^(1/3))^2 + x^(1/3) + 1) = (x^(1/3) - 1)(x^(2/3) + x^(1/3) + 1) = x + x^(2/3) + x^(1/3) - x^(2/3) - x^(1/3) - 1 = x - 1.

As for the fourth line of the solution, they remembered how to work with even-numbered exponents in polynomials. More specifically, they're attempting to find something to multiply (x^(1/4) - 1) by so that it will become (x - 1), since that would take care of the (x - 1) in the numerator and allow the limit to be taken. That's how they find the conjugate.

Hope that helps!