<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] 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 $-6$
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