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**relearning_calculus** 0. How do I make a cube root symbol?

the code is \sqrt[3] {x} to yield $\displaystyle \sqrt[3] {x}$

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1. What kind of sick person dreams up false proofs?

not necessarily sick. most of the time it is an extremely bored, smart person. but when it is a sick person, they tend to be the sickest, like beyond belief

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2. How is the hint supposed to help? If I transform the step $\displaystyle a^3 = -1$ to $\displaystyle a^3 + 1 = 0$, and from there to $\displaystyle (a^2 - a + 1)(a + 1) = 0$, I now have a more complicated expression -- one for which the solution is still -1 or +1, neither of which still play well in $\displaystyle a = 1 + a^2$.

we addressed the problem already, so i won't answer this now

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3. If I encountered this kind of problem without this helpful hint, how would I know to use it? It seems like a singular rabbit to pull out of a hat...

hehe, we didn't even use the hint. frankly, i don't see how it is helpful really.

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4. This is the big one... Following normal algebraic processes, starting from a reasonable assumption, this horrific result, uh, resulted. Is this a failure of algebra, or of the idiot user attempting to make it work?

definitely the latter