It took us thousands of years to properly define real numbers. High school students can live without a perfect explanation, or we can just teach limits before college since they are the fundamental concept if calculus.
Probably you were taught how to multiply irrational by the property of powers (a^b * c^b = (a*c)^b etc.).
You were not taught a grand unifying theory of multiplication, you were taught how to manipulate operations to turn them into more useful operations.
Teaching these laws also prepares you for when a and b are just symbolic reals with no structure and those laws are the only thing you can use to manipulate them.