Stage: 5 Challenge Level: 
The absurdity of it!
Find the exact values of $$\sqrt{2 + \sqrt 3}- \sqrt{2 - \sqrt3}$$ and of $$\root 3 \of {2 + \sqrt 5}+ \root 3 \of {2 - \sqrt 5}$$.
NOTES AND BACKGROUND
Take care with the algebra and the result comes out neatly.
Identities. Long problems. Pythagoras' theorem. Modulus arithmetic. Mathematical reasoning & proof. Quadratic equations. Powers & roots. Index notation/Indices. Surds. Golden ratio.
Published July 1999.
Copyright © 2007. University of Cambridge. All rights reserved.
NRICH is part of the family of activities in the Millennium Mathematics Project, which also includes the Plus and Motivate sites.
email: nrich@damtp.cam.ac.uk
