<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] Probably they are telling you a way to rationalize the denominator so you can do the sum.
$1=2-1=(\sqrt[3]{2}-\sqrt[3]{1})(\sqrt[3]{4}+\sqrt[3]{2}+\sqrt[3]{1})$
Similarly 
$1=3-2=(\sqrt[3]{3}-\sqrt[3]{2})(\sqrt[3]{9}+\sqrt[3]{6}+\sqrt[3]{4})$
and
$1=4-3=(\sqrt[3]{4}-\sqrt[3]{3})(\sqrt[3]{16}+\sqrt[3]{12}+\sqrt[3]{9})$
so you can multiply the first fraction by $\frac{\sqrt[3]{2}-\sqrt[3]{1}}{\sqrt[3]{2}-\sqrt[3]{1}}$, the second one by $\frac{\sqrt[3]{3}-\sqrt[3]{2}}{\sqrt[3]{3}-\sqrt[3]{2}}$ and the third one by $\frac{\sqrt[3]{4}-\sqrt[3]{3}}{\sqrt[3]{4}-\sqrt[3]{3}}$ and you get 
$$\frac{\sqrt[3]{2}-\sqrt[3]{1}}{2-1}+\frac{\sqrt[3]{3}-\sqrt[3]{2}}{3-2}+\frac{\sqrt[3]{4}-\sqrt[3]{3}}{4-3}=\sqrt[3]{2}-1+\sqrt[3]{3}-\sqrt[3]{2}+\sqrt[3]{4}-\sqrt[3]{3}=\sqrt[3]{4}-1$$
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