Try this beautiful problem from American Invitational Mathematics Examination, HANOI, 2018 based on Squares and square roots.
Let a=\((\sqrt2+\sqrt3+\sqrt6)(\sqrt2+\sqrt3-\sqrt6)(\sqrt3+\sqrt6-\sqrt2)(\sqrt6+\sqrt2-\sqrt3)\)
b=\((\sqrt2+\sqrt3+\sqrt5)(\sqrt2+\sqrt3-\sqrt5)(\sqrt3+\sqrt5-\sqrt2)(\sqrt5+\sqrt2-\sqrt3)\). The difference a-b belongs to the set
Algebra
Squares and square roots
Number Theory
Answer: is [-4,0).
HANOI, 2018
Elementary Number Theory by David Burton
(x+y+z)(x+y-z)(x-y+z)(-x+y+z)=2\((x^{2}y{2}+y^{2}z^{2}+z^{2}x^{2})-x^{4}-y^{4}-z^{4}\).
We get a-b=2(2+3)(6-5)-\(6^{2}+5^{2}\)=-1.
Then a-b belongs to [-4,0).