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 It's all in the roots! (Posted on 2004-06-04)
Simplify the following:
√(3 - √5) + √(4 + √7) + √(6 - √35)

 No Solution Yet Submitted by Purna Rating: 3.2857 (7 votes)

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 re(2): Long answer - very interesting | Comment 3 of 8 |
(In reply to re: Long answer - very interesting by Erik O.)

I got a similar result this way: I supposed there existed some identity like √(a±√b)= X±Y. Squaring to remove a square root sign, you get a±√b=X²+Y²±2XY.

Since this is ONE equation with TWO unknowns, and there is still another square root to remove, in order to simplify a little I decided to try out letting a=X²+Y² so some terms would go away.

That identity implies that X²=a/2+z and Y²=a/2-z. Substituting, ±√b=±2√(a²/4-z²). Squaring again to remove both square roots at the same time, we finally get z=(1/2)√(a²-b).

From now on, operating I reached the same √(14) result as e.g.

Edited on June 4, 2004, 2:59 pm
 Posted by Federico Kereki on 2004-06-04 14:54:09

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