Spanish Olympiad Regional Phase

Algebra Level 3

$\large \sqrt[3]{45 + 29\sqrt2} + \sqrt[3]{45-29\sqrt2} = \, ?$

1 solution

José Antonio Fuentes
Jan 20, 2020

$(3+\sqrt{2})^3=45+29\sqrt{2}$ and $(3-\sqrt{2})^3=45-29\sqrt{2}$ so $\sqrt[3]{45+29\sqrt{2}}+ \sqrt[3]{45-29\sqrt{2}}=3+\sqrt{2}+3-\sqrt{2}=6$ .

Peter Csorba - 1 year, 4 months ago

Yes, absolutely true, in the question is written root + root but in the answer I accidentally wrote root - root

José Antonio Fuentes - 1 year, 4 months ago

If you cube your version of $x$ correctly, you obtain $x^3 \; = \; 58\sqrt{2} -21x$ which has $x = 2\sqrt{2}$ as solution.

Mark Hennings - 1 year, 4 months ago

As I've said before, yes, you're right, in the question is written root + root but in the answer I accidentally wrote root - root

José Antonio Fuentes - 1 year, 4 months ago

Well, the question has been amended to +. It would seem sensible to amend your solution to match.

Mark Hennings - 1 year, 4 months ago