<s>[INST] simplify $\sqrt[3]{5+2\sqrt{13}}+\sqrt[3]{5-2\sqrt{13}}$ simplify $$\sqrt[3]{5+2\sqrt{13}}+\sqrt[3]{5-2\sqrt{13}}$$

1.$\frac{3}{2}$
2.$\frac{\sqrt[3]{65}}{4}$
3.$\sqrt[3]{2}$
4.$1$

I equal it to $\sqrt[3]{a}+\sqrt[3]{b}$ but I cant find $a$ and $b$
 [/INST] Let $X=\sqrt[3]{5+2\sqrt{13}}+\sqrt[3]{5-2\sqrt{13}}$ From $(A+B)^3=A^3+B^3+3AB(A+B)$ we get the equation $$X^3-3ABX-(A^3+B^3)=0$$
We have $(AB)^3=({5+2\sqrt{13}})({5-2\sqrt{13}})=-27\Rightarrow {AB}=-3$ and $A^3+B^3=10$. Hence our equation $$X^3+9X-10=0\iff (X-1)(X^2+X+10)=0$$
Thus $$X=1$$ It is the $4$ the asked simplification.
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