a = (1 + sqrt(2))
b = -sqrt(2)
then a and b are irrational numbers and...
a + b = (1 + sqrt(2)) + (-sqrt(2)) = 1 which is rational
a = sqrt(2)
b = 1/sqrt(2)
then a and b are both irrational and...
a*b = sqrt(2)*1/sqrt(2) = 1 which is rational
i should be using this oppurtunity to try out Latex, i don't know how to use it
EDIT: o sorry, i forgot a and b must be positive, so the first example is incorrect, i'll come up with another one
a = 10 - sqrt(5) + sqrt(2)
b = sqrt(5) - sqrt(2)
then a + b = 10 which is rational and both a and b are positive irrational numbers