I found this identity today in a test.
\[
\sqrt{\sqrt2+1} + \sqrt{\sqrt2-1} = \sqrt{2\sqrt2+2}
\]
It is interesting because it's of the somewhat rare form square root plus square root equals something simpler. Related to this is my post Rational sum of irrationals, where I come to the conclusion that the sum of two irrational square roots of fractions cannot be rational (that would make it simpler!), and also possibly my post on the fibonacci sequence, in which the sum of two exponentially growing irrational numbers equals an integer.
Related to this is a formula for the roots of a cubic polynomial. See MF85 by Wildberger at 22.00: https://www.youtube.com/watch?v=BQrig2uBsqs. Basically, a sum of two different cube roots of sums/differences of a natural number and a square root of a natural number can be an integer even though the cube roots are irrational.
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