√ (3+√(2√7+1)) - √ (3-√(2√7+1))
= √ ( (3+√(2√7+1)) - 2·√ ( (3+√(2√7+1))·(3-√(2√7+1)) ) + (3-√(2√7+1)) )
= √ ( 6 - 2·√ (9 - (2·√7 + 1)) )
= √ (6 - 2· √(8-2·√7) ) = √ (6 - 2· √(7-2·√7+1) )
= √ (6 - 2· √(√7-1)^2 ) = √ (6 - 2· (√7-1) )
= √(8-2·√7) = √(7-2·√7+1) = √(√7-1)^2
= √7-1