<s>[INST] Will the following expression be irrational, rational or integer? Will the following expression be irrational, rational or integer?
$$\sqrt[3]{\sqrt a +b} - \sqrt[3]{\sqrt a -b}$$
where $a$ = $52$ and $b$ = $5$ .
By intuition, I think this will be an integer. 
 [/INST] Let's use the identity $(\alpha + \beta)^3 = \alpha^3+\beta^3+3\alpha\beta(\alpha+\beta)$
Set $\alpha =\sqrt[3]{b+\sqrt{a}} \text{ and } \beta= \sqrt[3]{b - \sqrt{a}} \text { and } \alpha+\beta=x$
We know that $\alpha\beta = \sqrt[3]{b+\sqrt{a}} \times \sqrt[3]{b - \sqrt{a}} = \sqrt[3]{b^2-a} = \sqrt[3]{25-52} = -3$
So $x^3= b+\sqrt a + b-\sqrt a - 9x = 2b-9x = 10-9x$
You know that you are looking for a real answer, because $a$ is positive and the two cube roots are therefore cube roots of real numbers. Arturo has factorised the cubic, but it is easy to see that 1 is a root (integer roots must be divisors of 10).
</s>