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

how about

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