How to simplify $\sqrt[3]{29\sqrt{2}-45}-\sqrt[3]{29\sqrt{2}+45}$

Question

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?

Answer 1

HINT:

Let $$\sqrt[3]{29\sqrt{2}-45}-\sqrt[3]{29\sqrt{2}+45}=a$$

$$a^3=29\sqrt{2}-45-(29\sqrt{2}+45)-3a(-7)$$

as $(29\sqrt{2}-45)(29\sqrt{2}+45)=-7^3$

$$\iff a^3-21a+90=0$$ whose only real root is $-6$

Answer 2

Hint. $(3+\sqrt{2})^3=45+29\sqrt{2}.$

Answer 3

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.