Which statement is not always true?
a) the sum of a rational number and an irrational number is irrational.
b) the sum of two rational numbers is rational.
c) the product of two irrational numbers is irrational.
d) the product of two rational numbers is rational.

A. True. Summing any rational number with an irrational number leads to an irrational result. The proof is a bit lengthy so I'm leaving it out. 

B. True. Adding p/q with r/s leads to (ps+qr)/(qs) which is rational. Keep in mind that q and s cannot be zero.

C. False. One counter example is sqrt(3)*sqrt(12) = sqrt(3*12) = sqrt(36) = 6. This shows the product of two irrational numbers, in this case sqrt(3) and sqrt(12), multiplying to get a rational result 6 = 6/1.

D. True. Multiplying p/q and r/s leads to (p*r)/(q*s) which is rational. Keep in mind that q and s cannot be zero.



The final answer is choice C