Question

Why is the product of a rational number and an irrational number irrational

Answer 1

Think about a real example: $4* \sqrt{34}=4 \sqrt{34}$ and that is irrational.

Answer 2

The product of a rational and irrational number is not always irrational. For example, 0*pi=0 (a rational number). However, in cases where it does create an irrational number, it is because rational numbers are terminating, which means they do not go on forever. An irrational number does go on forever, so when combining something that will go on forever with something that will not, it will continue to go on forever.