<s>[INST] Can anyone explain why expressions of the form $\sqrt[3]{x-\sqrt{y}}+\sqrt[3]{x+\sqrt{y}}$ can be rational? $$\sqrt[3]{2-\sqrt{5}}+\sqrt[3]{2+\sqrt{5}}=1$$
Can anyone explain to me how this works? I don't understand why two irrational numbers cube rooted and added together return the number $1$.
I'm trying to find all the positive integer values for $x$ and $y$ in this formula
$$\sqrt[3]{x-\sqrt{y}}+\sqrt[3]{x+\sqrt{y}}=1$$
Sorry if this is a stupid question.
 [/INST] Note that
$$\sqrt[3]{2-\sqrt{5}}+\sqrt[3]{2+\sqrt{5}}=1$$
is a special case $(m=1)$ of the identity
$$s=\sqrt[3]{\left(3m-1\right) -\sqrt{ (8m-3)m^2}}
 + \sqrt[3]{\left(3m-1\right) +\sqrt{ (8m-3)m^2}} =1
$$
It can be verified by taking cubic power of both sides, i.e.
$$s^3-3(1-2m)s +2-6m=(s-1)(s^2+s+6m-2)=0$$
which leads to $s=1$. Then, write the identity in the form
$$\sqrt[3]{x-\sqrt{y}}+\sqrt[3]{x+\sqrt{y}}=1$$
$$x= 3m-1,\>\>\>\>\>y=(8m-3)m^2$$
$x$ and $y$ are both positive integer for $m=1,2,3...$. and there are infinite possibilities. The first few examples are listed below.
\begin{align}
\sqrt[3]{2-\sqrt{5}} + \sqrt[3]{2+\sqrt{5} } & =1 ,\>\>\>m=1\\
\sqrt[3]{5-\sqrt{52}} + \sqrt[3]{5+\sqrt{52} } & =1 ,\>\>\> m=2\\
\sqrt[3]{8-\sqrt{189}} + \sqrt[3]{8+\sqrt{189} } & = 1 ,\>\>\>m=3\\
 \sqrt[3]{11-\sqrt{464}} + \sqrt[3]{11+\sqrt{464} } & = 1 ,\>\>\>m=4\\
\sqrt[3]{14-\sqrt{925}} + \sqrt[3]{14+\sqrt{925} } & = 1 ,\>\>\>m=5\\
...&= 1,\>\>\>...
\end{align}
</s>