<s>[INST] Summing cube roots in fractions I found this problem, and understand the solution, but do not understand why they made the first assumption.
The problem:

The first line of the solution says that:

The cube root of $1$ plus the cube root of $2$ plus the cube root of $4$ is a factor of $2-1$.

Why are you meant to assume this to solve the problem?
 [/INST] They are pointing to the fact (it's not an assumption) that $$ (x^2+xy+y^2)(x-y) = x^3-y^3$$
that is
$$ \frac{1}{x^2+xy+y^2} = \frac{x-y}{x^3-y^3}$$
If you use that for $x=1$, $y=\sqrt[3]{2}$, you get
$$ \frac{1}{1+\sqrt[3]{2}+\sqrt[3]{4}} = \frac{\sqrt[3]{2}-1}{2-1} = \sqrt[3]{2}-1$$
To get the other two fractions use  $x=\sqrt[3]{2}$, $y=\sqrt[3]{3}$ and $x=\sqrt[3]{3}$, $y=\sqrt[3]{4}$.
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