prove or disprove the following statement.

Irrational numbers of the form a+√b are closed under multiplication where a and b are natural numbers

I don't think it's true

(2+√3)(5+√7) = 10+2√7+5√3+√21

I don't think that can be reduced to a+√b

Note, however, that a+b√D where D is fixed do form a quadratic field, which is closed under multiplication.