<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] Hint: Rewrite $x_{1,2}=5\pm 2\sqrt{13}$ in polar coorinates $x=|x|\exp(i\phi_{1,2})$. Determine $|x_{1,2}|=\sqrt{5^2+(2\sqrt{13})^2}$ and $\tan(\phi_{1,2})=\frac{\pm 2\sqrt{13}}{4}$
Use the periodicity of the exponential function in the complex domain
$\sqrt[3]{x_{1,2}}=|x_{1,2}|^{1/3}\exp(i\phi_{1,2}/3+2\pi k/3)$, in which $k \in \mathbb{Z}$ (you will have to check which range of values for $k$ give new solutions.
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