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

Let $a=\sqrt[3]{\sqrt{5}+2}$, $b=\sqrt[3]{\sqrt{5}-2}$ and $x=a-b$. Then,
$$x^3=\left(a-b\right)^3=a^3-3a^2b+3ab^2-b^3=a^3-b^3-3ab\left(a-b\right)\\=\sqrt{5}+2-\left(\sqrt{5}-2\right)-3x\sqrt[3]{\left(\sqrt{5}+2\right)\left(\sqrt{5}-2\right)}=4-3x$$
Then the answer is the real root of the equation $x^3+3x-4=0$,
$$x^3-3x-4=0\\\left(x-1\right)\left(x^2+x+4\right)=0\\\because x^2+x+4>0\\ \therefore x=1$$ We get $\sqrt[3]{\sqrt{5}+2}-\sqrt[3]{\sqrt{5}-2}=1$
