a rational number is a number that can be written on the form a/b where a and b are integers. we define multiplication out rational numbers to be"multiply the 2 denominator to get a new denominator and the 2 numerator to get a new number. a1/b1 * a2/b² = a3/b3. since multiplication it's closed on the integers, a3 and b3 are integers and be thus a3/b3 is rational by definition