<s>[INST] How to simplify $\sqrt[3]{29\sqrt{2}-45}-\sqrt[3]{29\sqrt{2}+45}$ I in trouble simplifying this:
$$\sqrt[3]{29\sqrt{2}-45}-\sqrt[3]{29\sqrt{2}+45}$$
couldn't find a solution. Can you help?
 [/INST] Those expressions usually come from Cardan's formula
$$
\sqrt[3]{\sqrt{\frac{q^2}{4}+\frac{p^3}{27}}-\frac{q}{2}}-
\sqrt[3]{\sqrt{\frac{q^2}{4}+\frac{p^3}{27}}+\frac{q}{2}}
$$
(see Wikipedia).
So we need $45=q/2$ or $q=90$ and so
$$
\frac{p^3}{27}+\frac{90^2}{4}=2\cdot29^2,
$$
that is,
$$
\frac{p^3}{27}=-343=-7^3
$$
which gives $p=-21$. Thus the number is a root of the equation
$$
x^3-21x+90=0
$$
and some attempts with the rational root test give $x=-6$ as the only real root.
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