<s>[INST] simplify $\sqrt{(45+4\sqrt{41})^3}-\sqrt{(45-4\sqrt{41})^3}$ simplify $\sqrt{(45+4\sqrt{41})^3}-\sqrt{(45-4\sqrt{41})^3}$.

1.$90^{\frac{3}{2}}$
2.$106\sqrt{41}$
3.$4\sqrt{41}$
4.$504$
5.$508$

My attempt:I do like this but I didn't get any of those five.
$\sqrt{(45+4\sqrt{41})^3}-\sqrt{(45-4\sqrt{41})^3}={\sqrt{45+4\sqrt{41}}}^3-\sqrt{45-4\sqrt{41}}^3$
$=(\sqrt{45+4\sqrt{41}}-\sqrt{45-4\sqrt{41}})(45+4\sqrt{41}+45-4\sqrt{41}+(\sqrt{45+4\sqrt{41}})(\sqrt{45-4\sqrt{41}})$
Now I do the nested radicals formula and I get $254\sqrt{41}$ which is none of those where did I mistaked?
 [/INST] Is this the nested radical formula?
$\sqrt{45 \pm 4\sqrt{41}} = a \pm b\sqrt{41}$
$45 \pm 4\sqrt{41} = (a^2 + 41b^2) \pm 2ab\sqrt{41}$
$a^2 + 41b^ = 45; 2ab = 4 \implies a=2;b = 1$
So $\sqrt{45 \pm 4\sqrt{41}} = |2 \pm \sqrt{41}|= \pm 2 + \sqrt{41}$
Plugging that into: $=(\sqrt{45+4\sqrt{41}}-\sqrt{45-4\sqrt{41}})(45+4\sqrt{41}+45-4\sqrt{41}+(\sqrt{45+4\sqrt{41}})(\sqrt{45-4\sqrt{41}}))$
We get $(2+\sqrt{41} - (-2+\sqrt{41}))(90+(2 + \sqrt{41})(-2+\sqrt{41}))=$
$4(90 + (-4 + 41)) = 508$
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It would have been easier not to do all the expanding:
$(2 + \sqrt{41})^3 - (-2+\sqrt{41})^3=$
$(2^3 + 3*2^2*\sqrt{41}  + 3*2*\sqrt{41}^2+\sqrt{41}^3)- (-2^3 + 3*2^2*\sqrt{41}  - 3*2*\sqrt{41}^2+\sqrt{41}^3)=$
$2(8 + 6*41) = 508$.
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