<s>[INST] Prove that $(5^{\frac 12} + 2)^{\frac 13}- (5^{\frac 12} - 2)^{\frac 13}$ is an integer and find its value I had proceed this question by taking
$$x =(5^{\frac 12} + 2)^{\frac 13}- (5^{\frac 12} - 2)^{\frac 13}$$
Then
$$x + (5^{\frac 12} -2)^{\frac 13} =(5^{\frac 12} + 2)^{\frac 13}$$
And then cubing both sides and then solving for $x$ by using Cardano's method but I got the same equation at last that is
$$x =(5^{\frac 12} + 2)^{\frac 13}- (5^{\frac 12} - 2)^{\frac 13}$$
Now I don't know how to solve this
 [/INST] Let $$x=\sqrt[3]{\sqrt5+2}-\sqrt[3]{\sqrt5-2},$$ 
raise it to the third power:
$$\begin{align}
x^3&=\sqrt5+2-3\sqrt[3]{\sqrt5+2}^2\sqrt[3]{\sqrt5-2}
+3\sqrt[3]{\sqrt5+2}\sqrt[3]{\sqrt5-2}^2-(\sqrt5-2)\\
&=4-3\cdot\sqrt[3]{\sqrt5+2}\sqrt[3]{\sqrt5-2}\cdot\bigl(\sqrt[3]{\sqrt5+2}-\sqrt[3]{\sqrt5-2}\bigr)\\
&=4-3\cdot\sqrt[3]{5-4}\cdot x\\
&=4-3\cdot1\cdot x.
\end{align}
$$
Now show that $x=1$ is the only solution of $x^3+3x-4=0$.
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